[From Bruce
Abbott (2018.08.28.1800 EDT)]
Â
[Martin Taylor 2018.08.28.13.45]
···
I think I’m going to rest my case.Â
I simply don’t know what else I can say to make my position
clear.
Bruce
[From Bruce Abbott (2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
[From Bruce Abbott (2018.08.28.1800 EDT)]
[Martin Taylor 2018.08.28.13.45]
It’s quite clear that my responses to you lead you to perceive that I misunderstand you, and I can assure you that your responses to me miss the point of many of my comments. I believe that you have a preconception of what I want to say, and interpret the inevitable ambiguities in my writing in that context. I presume I am doing the same with respect to your comments.
You say:
I think I’m going to rest my case. I simply don’t know what else I can say to make my position clear.
Bruce
I empathise, since I feel the same way. Indeed, a couple of days ago I thought your position was quite clear, and I agreed with you, but then Rupert made a point that I felt was equally cogent, so I tried to go back to first principles and see If I could address both positions and decide for myself. The result of that was that I disagreed with the position I thought had been clear in your writing, and I explained why in some detail. But your response to that ignored or mis-stated some reasoning I thought critical, while at the same time making me unsure of what your position actually is. Your comments that I have excised make me even less sure what you believe. What seemed clear as crystal is now clear as mud in my mind.
But for Rupert, halting the conversation leaves his initial question open. So I would rather not let the question lapse.
Let me take just one point early in your comments. If we agree on this, maybe we can move on. If we disagree here, then we should find out what underlying (mis)understanding causes the disagreement. Maybe it will help to make the rest of your comments clearer to me.
[Martin Taylor 2018.08.27.14.34]
MT: I found myself in the strange position of having reasons to accept both sides of an opposition. On the one side, following Bruce, is a notion that after the wheels hit the rocks a permanent offset is added to the output, but its effect is reduced by the control gain, while on the other, following Rupert, is that it is obvious that if the driver took the steering wheel off the post and put it back at some random angle relative to where it was, that would make no difference at all to the ability of the driver to keep to the middle of the lane. So why should the misalignment caused by the rocks make any difference? It has the same relation to the angle of the steering wheel as does taking the wheel off and putting it back differently.
BA: As I noted to Rupert, a human driver would have no problem in either case, but the standard PCT controller with a leaky integrator output function would set the steering wheel position to the former “straight aheadâ€? position, and this would result in the car turning. The control system would compensate by turning the steering wheel in the opposite direction. In the misalignment case this would rebalance the steering forces so that the car would track straight ahead – a net zero steering angle. In the steeering wheel resetting case this would change the steering wheel position so that the steering angle is zero.
After the car had been steered away from the lane centre, why would the effect of the PCT controller be to compensate so as to make the car steer straight ahead? Why did you even mention “straight ahead” if you did not mean to imply that heading angle was a controlled variable, as you say you did not? What is special about a “straight ahead” or “net zero steering angle” when the PIL controller has non-zero error?
I would expect the PCT PIL controller to adjust its output to the steering wheel so as to make the heading come around to an angle that was not straight ahead, but rather an angle that would lead to a reducing PIL error value. Do you disagree?
No, I agree. The steering angle will be adjusted so as to make the heading come around to an angle that will reduce the PIL error value as the car continues to move forward.
Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference? What will that position be? What will the car’s heading be, relative to the road’s heading?
Bruce
Martin
[From Bruce
Abbott (2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
…
···
I
would expect the PCT PIL controller to adjust its output to
the steering wheel so as to make the heading come around to an
angle that was not straight ahead, but rather an angle that
would lead to a reducing PIL error value. Do you disagree?
No, I agree. The steering angle
will be adjusted so as to make the heading come around to an
angle that will reduce the PIL error value as the car
continues to move forward.
Eventually, if the controller is
sufficiently damped, the car’s position will stabilize
relative to the reference position (lane center). Assuming
that the road is straight, what will the car’s steering
angle be, once the car has reached its asymptotic position
relative to the reference? What will that position be?
What will the car’s heading be, relative to the road’s
heading?
[From Bruce Abbott (2018.08.29.1210 EDT)]
[Martin Taylor 2018.08.29.08.50]
[From Bruce Abbott (2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
…
MT: I would expect the PCT PIL controller to adjust its output to the steering wheel so as to make the heading come around to an angle that was not straight ahead, but rather an angle that would lead to a reducing PIL error value. Do you disagree?
BA: No, I agree. The steering angle will be adjusted so as to make the heading come around to an angle that will reduce the PIL error value as the car continues to move forward.
BA: Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference? What will that position be? What will the car’s heading be, relative to the road’s heading?
MT: The word “asymptotic” is crucial in your question. It implies a target value that a variable will approach ever more slowly and will reach exactly only after infinite time. Last question first, answers referring to a “standard” PCT loop:
Theoretically infinite, but in our simulations and the real world, in finite time and usually fairly quickly. The simulations do because there are limits to the precision with which the computations are carried out. In the real world there are factors such as damping that have the same effect. (Of course, these can also be added to the simulation if desired.)
Q: “What will the car’s heading be, relative to the road’s heading?”
···
A: The car will be heading parallel to the road’s heading.
Yes. Does this fact imply that heading is a controlled variable?
Q: “Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference?”
A: There is no way to determine the angle of the steering wheel, but the road wheels will be directed along the car’s axis, a direction we can conveniently call zero.
Yes. If they were not, how would this affect the car’s position relative to the reference position?
Comment 1: The road wheel angle determines only the second derivative of the car’s position relative to the lane centre, but asymptotically, for any variable that does approach an asymptote, all derivatives of every order must approach zero.
Comment 2: “Sufficiently damped” is an unnecessary modifier. “Stable” is the required criterion, because the only options are stable and unstable. In the example situation, the other two dynamical possibilities (“chaotic” and “orbital”) are either unattainable or of probability approaching zero.
I was thinking in terms of “underdamped,â€? “overdamped, and “critically damped.â€? An underdamped control system will oscillate, perhaps with increasing or decreasing amplitude over cycles. An overdamped system approaches its asymptote more slowly than necessary for stability—it is relatively sluggish. A critically damped system hass the minimum damping necessary to prevent oscillation.
Q: “…once the car has reached its asymptotic position relative to the reference? What will that position be?”
A: This is a question that hides a subtlety. I start with the obvious.
Ignoring the implication of "once" (that the car would at some moment reach its asymptotic position), the answer is that for a standard PCT controller with a leaky integrator of gain rate g and a leak rate r, the asymptotic position is r*X/g where X is the amount by which the car is distant from the reference position when control is re-established after the change of relationship between the steering wheel angle and the road wheel angle.
I’m not sure I follow this. Leakage in the PCT output function applies to the change in output. If the car is steering a straight line parallel to the reference position, the error is constant and there is no change in output. Consequently, in the steady state shouldn’t the leakage constant disappear from the loop gain computation?
"*After the change in relationship*" requires thinking about the dynamics of the system, because it implies that there exists non-zero loop transport lag, and during that lag if the road wheel angle was changed by the event, the car will have veered some distance off course. If the lag were zero, the deviation also would be zero, and the car would not have deviated at all, unless the rocks that cause the misalignment also, and independently nd not mentioned in the problem statement, bumped the car sideways the distance X. If the misalignment is caused by removal and replacement of the steering wheel while the car was moving straight ahead, X would be zero.
Assume that the car remains at its reference position after the change. The controller’s output function enforces that a certain steering wheel position will produce “straight aheadâ€? driving when there is no error in the car’s position on the road.Â
Scenario 1: If the steering wheel position is changed while the controller is holding the steering wheel at the position that will produce straight ahead driving, then the change in steering wheel position can only be effected by rotating the steering angle of the wheels. With the steering angle no longer zero (straight ahead), the car will begin to turn. The controller now experiences a non-zero error and in response will turn the steering wheel in the direction of the reference position. This will return the steering angle to zero and the car will track straight down the road, but offset enough that the steering wheel remains turned enough to keep the steering angle at zero.
Scenario 2: If the controller is disconnected from the steering wheel, the steering wheel taken off, rotated, and replaced, and the controller reconnected so that the new steering wheel position produces the same steering angle as before the rotation, there will be no effect on the car’s steering angle and therefore no effect on the car’s position. I believe that you have been assuming Scenario 2, whereas I have been assuming Scenario 1.
Now assume instead that the rocks upset the steering alignment but again do not change the car’s position or heading while it is at its reference position and heading straight down the road. The steering misalignment will cause the car to start veering to the left, creating a position error. The controller will respond to this error by turning the wheels toward the right, in the direction of the reference position. The system will stabilize at the position at which the error is great enough to maintain a steering angle that prevents the error from growing further. This is the angle at which the net steering angle is again zero, although neither front wheel is pointing straight ahead. The car will continue to track parallel to the road, but at an offset relative to reference.
The answer to your question is therefore "*if the loop transport lag is zero, the misalignment will not have any effect on the car's position relative to the lane centre. If the loop transport lag is non-zero, the ratio r/g cannot be less than some value 1/G that is on the verge of causing the loop to become unstable, and the asymptotic position will be X/G where X is the deviation caused by the event before control is regained. Additionally, even if the car is bumped sideways by the rocks that caused the misalignment, if the loop transport lag is zero, G can be infinitely large. This would mean that the event causes an infinitesimally short deviation from the original position.*"
Do you agree?
No, see above.  After the system stabilizes, the car will maintain a position parallel to the road but offset from the reference position. A nice illustration of this effect is presented in Bill Powers’ 1979 Byte series, June issue.
Bruce
[From Bruce
Abbott (2018.08.29.1210 EDT)]
···
[From Bruce Abbott(2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
…
MT: I would expect the
PCT PIL controller to adjust its output to the steering
wheel so as to make the heading come around to an angle that
was not straight ahead, but rather an angle that would lead
to a reducing PIL error value. Do you disagree?
BA: No, I agree. The steeringangle will be adjusted so as to make the heading come
around to an angle that will reduce the PIL error value as
the car continues to move forward.
BA: Eventually, if the controlleris sufficiently damped, the car’s position will stabilize
relative to the reference position (lane center).Â
Assuming that the road is straight, what will the car’s
steering angle be, once the car has reached its asymptotic
position relative to the reference? What will that
position be? What will the car’s heading be, relative to
the road’s heading?
Â
MT: The word “asymptotic”
is crucial in your question. It implies a target value that a
variable will approach ever more slowly and will reach exactly
only after infinite time. Last question first, answers
referring to a “standard” PCT loop:
Â
Theoretically
infinite, but in our simulations and the real world, in
finite time and usually fairly quickly. The simulations do
because there are limits to the precision with which the
computations are carried out.
[Martin Taylor 2018.08.28.20.55]
I
would expect the PCT PIL controller to adjust its output to
the steering wheel so as to make the heading come around to an
angle that was not straight ahead, but rather an angle that
would lead to a reducing PIL error value. Do you disagree?
No, I agree. The steering angle
will be adjusted so as to make the heading come around to an
angle that will reduce the PIL error value as the car
continues to move forward.
Eventually, if the controller is
sufficiently damped, the car’s position will stabilize
relative to the reference position (lane center). Assuming
that the road is straight, what will the car’s steering
angle be, once the car has reached its asymptotic position
relative to the reference?
[From Bruce Abbott (2018.08.29.1750 EDT)]
[Martin Taylor 2018.08.29.12.54]
[From Bruce Abbott (2018.08.29.1210 EDT)]
This is good. We come to a place where we interpret the PCT controller differently, so we have a possible opportunity for reconciliation. This time, I will not excise any of the earlier text.
[Martin Taylor 2018.08.29.08.50]
[From Bruce Abbott (2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
…
MT: I would expect the PCT PIL controller to adjust its output to the steering wheel so as to make the heading come around to an angle that was not straight ahead, but rather an angle that would lead to a reducing PIL error value. Do you disagree?
BA: No, I agree. The steering angle will be adjusted so as to make the heading come around to an angle that will reduce the PIL error value as the car continues to move forward.
BA: Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference? What will that position be? What will the car’s heading be, relative to the road’s heading?
MT: The word “asymptotic” is crucial in your question. It implies a target value that a variable will approach ever more slowly and will reach exactly only after infinite time. Last question first, answers referring to a “standard” PCT loop:
BA: Theoretically infinite, but in our simulations and the real world, in finite time and usually fairly quickly. The simulations do because there are limits to the precision with which the computations are carried out.
Right. In the analogue world there is also a limit to precision, imposed by system noise, whether in neural firing rates or in environmental irregularities. In vision it is why we need glasses, and even with them we cannot with the naked eye resolve or even see Jupiter’s Galilean moons. In control modelling, we often coalesce all these precision limits into one, which we embed in a tolerance zone, a zone within which deviation from a reference value (also noisy in the real world) does not produce a non-zero error value. (That’s not the only reason for a tolerance zone, but it is a sufficient reason.)
BA: In the real world there are factors such as damping that have the same effect.
No, they have a very different effect. Damping refers to the effect of energy losses from the system by friction or viscosity. Even as you seem to use the word (see below), it has little or no connection with precision, either of perception or of action.
I should have clarified there. By “same effect� I meant that the system will stabilize in reasonable time rather than taking an infinite time to reach asymptote. I did not mean that limits to precision and damping do this in the same way.
(Of course, these can also be added to the simulation if desired.)
Q: “What will the car’s heading be, relative to the road’s heading?”
A: The car will be heading parallel to the road’s heading.
Yes. Does this fact imply that heading is a controlled variable?
No. In the problem situation, heading cannot be controlled independently of the position-in-lane (PIL) variable, though it could very well be controlled as a lower-level variable with a reference value set by the PIL control unit. Even the latter depends on how you model the control hierarchy and since we are agnostic about that and are considering only the PIL control loop, the actual heading must be treated as a side-effect variable.
Yes, a side effect. That was my thinking from the beginning but somehow at some point in our exchanges you thought I was implying that heading was a controlled variable.
Q: “Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference?”
A: There is no way to determine the angle of the steering wheel, but the road wheels will be directed along the car’s axis, a direction we can conveniently call zero.
BA: Yes. If they were not, how would this affect the car’s position relative to the reference position?
How would what affect the car’s position relative to the reference position? The steering wheel angle?
You said “There is no way to determine the angle of the steering wheel, but the road wheels will be directed along the car’s axis . . .� I answered if they were not . . .� As there is only one steering wheel but two road wheels, why would you think that I’m asking about the steering wheel?
That can’t do it, because after the event, a totally different steering wheel angle corresponds to a zero of the second derivative of the PIL. What we can say is that the first derivative of the steering wheel angle, no matter what the actual angle, affects the third derivative of the PIL in a way unchanged by the event. The actual steering wheel angle is something an external observer can see to have changed, but that doesn’t affect the operation of th PIL control loop in the slightest.
But it does affect the relation between the position of the steering wheel and the angle of the wheels. If the controller’s output function sets the steering wheel to a specific angle when the positional error is zero, that same steering wheel angle will no longer keep the car driving straight ahead when the car is at the reference position and running parallel to the lane.
Comment 1: The road wheel angle determines only the second derivative of the car’s position relative to the lane centre, but asymptotically, for any variable that does approach an asymptote, all derivatives of every order must approach zero.
Yes, the road wheel angle determines only the second derivative of the car’s position. The first derivative is the rate of drift, which depends on the heading. Both are zero when the car is steering straight and parallel to the lane center.
Comment 2: “Sufficiently damped” is an unnecessary modifier. “Stable” is the required criterion, because the only options are stable and unstable. In the example situation, the other two dynamical possibilities (“chaotic” and “orbital”) are either unattainable or of probability approaching zero.
I was thinking in terms of “underdamped,â€? “overdamped, and “critically damped.â€? An underdamped control system will oscillate, perhaps with increasing or decreasing amplitude over cycles. An overdamped system approaches its asymptote more slowly than necessary for stability—it iis relatively sluggish. A critically damped system has the minimum damping necessary to prevent oscillation.
I suppose we must mean something different by “damping”. As I use the word, there is no damping in any of the models we typically use. The only available parameters in the simple PCT model that is usually fitted are gain rate, leak rate, and transport lag. The interactions among these determine the dynamical properties of the loop that you ascribe to different degrees of “damping”. I defined what I think of as “damping” above. Could you define how you use the word part from using it as a description of the dynamical properties of the loop as you did here?
See https://en.wikipedia.org/wiki/Damping_ratio . Â If you fiddle with the loop gains of our standard PCT models, too large a gain will drive them into oscillation, and too little will produce sluggish operation and a relatively large deviation from reference in the steady state when there is a constant disturbance acting on the CV.
Q: “…once the car has reached its asymptotic position relative to the reference? What will that position be?”
A: This is a question that hides a subtlety. I start with the obvious.
Ignoring the implication of "once" (that the car would at some moment reach its asymptotic position), the answer is that for a standard PCT controller with a leaky integrator of gain rate g and a leak rate r, the asymptotic position is r*X/g where X is the amount by which the car is distant from the reference position when control is re-established after the change of relationship between the steering wheel angle and the road wheel angle.
BA: I’m not sure I follow this. Leakage in the PCT output function applies to the change in output. If the car is steering a straight line parallel to the reference position, the error is constant and there is no change in output. Consequently, in the steady state shouldn’t the leakage constant disappear from the loop gain computation?
The structure of the loop doesn’t depend on the values of the variables represented by the signals that pass round the loop. If the error is constant but there is a steady output, there will be a steady leak. In a sampled version, expanded to show the individual physical terms, we write O(t) = O(t-1) + ge(t) - rO(t-1), where g and r are per-sample rates.
That equation can be solved for O(t) = O(t-1) and e(t) = e(t-1):
O(t) = (1-r)O(t-1) + ge(t) = (1-r)(O(t) + ge(t).
rO(t) = ge(t)
O(t) = (g/r)e(t) = G*e(t).
Or, the steady state error is 1/G times the disturbance that is exactly opposed by the steady state output, just as we get by following around the loop in the usual way, if we attribute all the loop gain the the output function.
Let’s see what the loop gain is in terms of g and r, assuming that the dynamic gain of the rest of the loop is minus unity, as it is in many (most) simulations. We want to find dO(t)/de(t) the relative rates of change of O and e. We have
dO(t)/dt = (1-r)dO(t-1)/dt + gde(t)/dt
Assuming close sampling, dO(t)/dt ≈ dO(t-1)/dt
dO(t)/dt = (1-r)dO(t)/dt + gde(t)/dt
rdO(t)/dt = gde(t)/dt
dO(t)/dt = (g/r)de(t)/dt = Gde(t)/dt,
So the dynamical loop gain is the same as the steady-state loop gain (which is hardly a coincidence, though with nonlinear systems this might not be the case.
O.K., got it!
"*After the change in relationship*" requires thinking about the dynamics of the system, because it implies that there exists non-zero loop transport lag, and during that lag if the road wheel angle was changed by the event, the car will have veered some distance off course. If the lag were zero, the deviation also would be zero, and the car would not have deviated at all, unless the rocks that cause the misalignment also, and independently nd not mentioned in the problem statement, bumped the car sideways the distance X. If the misalignment is caused by removal and replacement of the steering wheel while the car was moving straight ahead, X would be zero.
BA: Assume that the car remains at its reference position after the change. The controller’s output function enforces that a certain steering wheel position will produce “straight ahead� driving when there is no error in the car’s position on the road.
For sure. But the earlier point was that the value of this angle is unknowable. That it exists is necessarily true.
BA: Scenario 1: If the steering wheel position is changed while the controller is holding the steering wheel at the position that will produce straight ahead driving, then the change in steering wheel position can only be effected by rotating the steering angle of the wheels. With the steering angle no longer zero (straight ahead), the car will begin to turn.
Yes, but in the absence of transport lag, the integrated effect of this “begin” is zero, because it is instantly corrected. Transport lag (including integration lag) is what allows “begin to turn” to become …
I do not understand your purpose in bringing a case in which transport lag is absent. No real system works this way.
BA: The controller now experiences a non-zero error and in response will turn the steering wheel in the direction of the reference position. This will return the steering angle to zero and the car will track straight down the road, but offset enough that the steering wheel remains turned enough to keep the steering angle at zero.
Yes, if g/r is finite, meaning that r > 0. The offset will asymptote at r/g times the maximum deviation of the PIL during the transport lag.
Again, I fail to understand why you are bring transport lag into the picture. Over iterations the offset will grow until it produces a zero steering angle. In the above, I interpret you to be saying that the asymptotic offset will depend on how much deviation develops during the time required to complete one iteration. As I understand it, the offset will depend on the loop gain.
BA: Scenario 2: If the controller is disconnected from the steering wheel, the steering wheel taken off, rotated, and replaced, and the controller reconnected so that the new steering wheel position produces the same steering angle as before the rotation, there will be no effect on the car’s steering angle and therefore no effect on the car’s position. I believe that you have been assuming Scenario 2, whereas I have been assuming Scenario 1.
Not true. I have been assuming both scenarios and I believed I had illustrated the difference between them and why transport lag matters. Apparently I didn’t, but maybe the development above will do the trick.
No, sorry, I’m still not following you. Not only am I mystified as to why transport lag matters with respect to the offset from reference that will exist at asymptote, I can see from simulation that such an offset does occur when there is a constant disturbance acting on the CV, and that its magnitude depends on loop gain.
BA: Now assume instead that the rocks upset the steering alignment but again do not change the car’s position or heading while it is at its reference position and heading straight down the road. The steering misalignment will cause the car to start veering to the left, creating a position error. The controller will respond to this error by turning the wheels toward the right, in the direction of the reference position. The system will stabilize at the position at which the error is great enough to maintain a steering angle that prevents the error from growing further. This is the angle at which the net steering angle is again zero, although neither front wheel is pointing straight ahead. The car will continue to track parallel to the road, but at an offset relative to reference.
Agreed. But to belabour the point, please do keep in mind the difference between “to start veering to the left” and “is offset to the left”. The latter requires integration of the amount of veering over time, that time being the transport lag in the loop. Â
O.K., but throughout this discussion I’ve been dealing with the effect of the steering misalignment on the steady-state position of the car relative to the reference position.  I have never assumed that the system will start doing anything instantly.  I am aware that nothing in the loop takes place instantly, and that transport lags produce phase shifts that can adversely affect performance. To me it seemed that you were saying that the system would track the reference with zero error, for reasons having something (incomprehensible to me) having to do with transport lag.  In fact, it still seems that you are saying that with respect to your steering wheel resetting scenario, whereas my conclusion is that a steady-state offset will develop in either case.
MT: The answer to your question is therefore “if the loop transport lag is zero, the misalignment will not have any effect on the car’s position relative to the lane centre. If the loop transport lag is non-zero, the ratio r/g cannot be less than some value 1/G that is on the verge of causing the loop to become unstable, and the asymptotic position will be X/G where X is the deviation caused by the event before control is regained. Additionally, even if the car is bumped sideways by the rocks that caused the misalignment, if the loop transport lag is zero, G can be infinitely large. This would mean that the event causes an infinitesimally short deviation from the original position.”
MT: Do you agree?
BA: No, see above. After the system stabilizes, the car will maintain a position parallel to the road but offset from the reference position.
Do you now agree with my italicised comment? If not, what about my preceding comments creates a problem? I assume that you agree that too high a value of G can lead to instability, that this critical value is higher, the shorter the transport lag, and that an infinite value of G would instantly and completely correct any PIL error. If you still disagree with the italicised comment, I think you must disagree with one of these statements. If so, which, and if not, what have I missed?
Still unclear about the zero transport lag and infinite G situations, including why you bring them up as no system has zero lag and/or infinite loop gain. Weren’t we talking about the simulation in the video? Secondly, I don’t understand why you expect the deviation “caused by the event before control is regained� to affect the magnitude of the deviation after the system stabilizes. I’m starting to see how, by your view, the transport lag would lead to different deviations when it is zero versus nonzero: With zero transport lag there is no time for a deviation to develop, otherwise, there is. Or maybe I’m confused about that, too.
When G is finite, I do agree with your “after the system stabilizes”, as I hope I made clear above.
Except we still disagree about what that offset depends on. I say it depends on the size of error that will produce an output that will keep the net steering angle at zero and thereby prevent any further increase in offset. You seem to be saying that it depends on how much deviation develops “during the transport lag� “before control is regained.� But control is regained only when, as I stated, the error has grown large enough to produce a disturbance-compensating output, i.e., one that sets the steering angle at zero. That is not a function of how long events take to circulate around the loop.  Are we somehow saying the same thing? I’m totally confused.
BA: A nice illustration of this effect is presented in Bill Powers’ 1979 Byte series, June issue.
Note the following quote from that article: “First, the feedback function is essentially an integrator, and so puts a lag into the control process. This alone would not cause a problem, but Chip also contains a transport lag; he cannot actually produce an output at’ the same instant that the input occurs, …”
Did you notice the steady-state offset that is proportional to the strength of the sidewind?
Bruce
[From Bruce Abbott (2018.08.29.1915 EDT)]
[From Rupert Young (2018.08.29 22.10)]
(Bruce Abbott (2018.08.29.0835 EDT)]
[Martin Taylor 2018.08.28.20.55]
I would expect the PCT PIL controller to adjust its output to the steering wheel so as to make the heading come around to an angle that was not straight ahead, but rather an angle that would lead to a reducing PIL error value. Do you disagree?
BA: No, I agree. The steering angle will be adjusted so as to make the heading come around to an angle that will reduce the PIL error value as the car continues to move forward.
BA: Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference?
Do you mean the angle of the steering wheel? It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
No, the “steering angle� is the angle of the car’s front wheels relative to its body. Straight ahead = zero steering angle, or parallel to the long axis of the car’s body. In the absence of disturbances, a zero steering angle has the car moving in a straight line. Steering angle is determined by the output of the position control system in the PID example of the video, so we can know it exactly. It’s determined by the steering wheel position in real autos.
BA: What will that position be?
It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
It’s irrelevant? That’s what the position control system is designed to control!
BA: What will the car’s heading be, relative to the road’s heading?
It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
No, when the car’s position control system is functioning properly, the car’s heading will be parallel to the road (in the absence of disturbances). With disturbances such as a crosswind added, there will be some variation in heading because the controller will be changing the steering angle as necessary to compensate for them, and given system lags there will be some variation because the disturbances can’t be compensated for instantaneously. However, unless the disturbances overwhelm the system’s ability to compensate, the average heading will remain parallel to the road.
However, in all cases, if the driver is controlling her perception of the position of the car then that will match her reference.
No, not at all times.  Disturbances will cause her car’s position to deviate from her reference position until she can compensate for their effect.  But what about a disturbance consisting of a constant crosswind? She will be able to compensate for the crosswind only by keeping the car’s steering angle turned against the wind. If her control system is a proportional one, so that the steering angle she uses depends strictly on the size of the deviation (error) between her sensed position and reference position, she will be able to maintain the proper compensatory steering angle only by maintaining the proper offset from the reference. Thus, if the crosswind is pushing her car to the right, she will have to keep her car to the right of reference far enough to produce the required compensatory steering angle to the left.
Of course, real drivers do not appear to behave this way. There are two possible explanations. One, they do maintain such an offset, but it is so small as to go unnoticed. This implies that the system has fairly high loop gain. Two, their position control system is not structured as a pure proportional control system. Either there is an added feature (such as integral control as in the PID controller), or the system is organized differently.  For example, it might turn out to be a hierarchical control system with the position control system setting the reference for a lateral velocity control system, which in acts to control lateral velocity by varying the steering angle. Another possibility is that the driver compensates for the offset by resetting the reference by the amount of the offset. The would allow the car to follow the “real� reference (where the driver wants the car to be) by in effect subtracting out the offset.
Bruce
Â
···
BA: No, I agree. The steeringangle will be adjusted so as to make the heading come
around to an angle that will reduce the PIL error value
as the car continues to move forward.
BA: Eventually, if thecontroller is sufficiently damped, the car’s position
will stabilize relative to the reference position (lane
center). Assuming that the road is straight, what will
the car’s steering angle be, once the car has reached
its asymptotic position relative to the reference?Â
Do you mean the angle of the steering wheel? It's irrelevant.
We can’t know it as it depends on the environment and
disturbances.
Â
No, the
“steering angle� is the angle of the car’s front wheels
relative to its body. Straight ahead = zero steering angle,
or parallel to the long axis of the car’s body. In the
absence of disturbances, a zero steering angle has the car
moving in a straight line. Steering angle is determined by
the output of the position control system in the PID example
of the video, so we can know it exactly. It’s determined by
the steering wheel position in real autos.
BA: What will that positionbe?Â
It's irrelevant. We can't know it as it depends on the
environment and disturbances.
Â
It’s
irrelevant? That’s what the position control system is
designed to control!
BA: What will the car’s headingbe, relative to the road’s heading?
It's irrelevant. We can't know it as it depends on the
environment and disturbances.
No, when the car’s position control
system is functioning properly, the car’s heading will be
parallel to the road (in the absence of disturbances). With
disturbances such as a crosswind added, there will be some
variation in heading because the controller will be changing
the steering angle as necessary to compensate for them, and
given system lags there will be some variation because the
disturbances can’t be compensated for instantaneously.
[From Bruce Abbott (2018.09.02.1125 EDT)]
[From Rupert Young (2018.09.01 13.15)]
(Bruce Abbott (2018.08.29.1915 EDT)]
BA: No, I agree. The steering angle will be adjusted so as to make the heading come around to an angle that will reduce the PIL error value as the car continues to move forward.
BA: Eventually, if the controller is sufficiently damped, the car’s position will stabilize relative to the reference position (lane center). Assuming that the road is straight, what will the car’s steering angle be, once the car has reached its asymptotic position relative to the reference?
RY: Do you mean the angle of the steering wheel? It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
BA: No, the “steering angle� is the angle of the car’s front wheels relative to its body. Straight ahead = zero steering angle, or parallel to the long axis of the car’s body. In the absence of disturbances, a zero steering angle has the car moving in a straight line. Steering angle is determined by the output of the position control system in the PID example of the video, so we can know it exactly. It’s determined by the steering wheel position in real autos.
RY: Ok, but that’s irrelevant too. There is no such thing as “Straight ahead” with respect to the steering angle in the real world, it depends on the environment. But that doesn’t matter as it is perception we are controlling not physical values of the world.
BA: What will that position be?
RY: It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
BA: It’s irrelevant? That’s what the position control system is designed to control!
RY: What’s being controlled is the perception of the position from the point of view of the human driver, which is probably what you mean. The position (physical centre) of the car is unlikely to be the same. We sit to one side of the car so even though our perceptual error will be zero the position of the car will be offset from the centre of the lane (and there will be different offsets with different cars). But that doesn’t matter as it is perception we are controlling not physical values of the world.
Actually, I was thinking in terms of the car in the video, not the human case with all these considerations such as the driver’s view of the road being brought into the picture. It’s not necessary and would only add confusion when working out how a proportional control system behaves.
BA: What will the car’s heading be, relative to the road’s heading?
RY: It’s irrelevant. We can’t know it as it depends on the environment and disturbances.
BA: No, when the car’s position control system is functioning properly, the car’s heading will be parallel to the road (in the absence of disturbances). With disturbances such as a crosswind added, there will be some variation in heading because the controller will be changing the steering angle as necessary to compensate for them, and given system lags there will be some variation because the disturbances can’t be compensated for instantaneously.
RY: Even though you say no, you are agreeing with what I said. It depends on the environment and disturbances. Also if the wheels are not perfectly aligned with the chassis of the car (which will always be the case in reality) then when the wheels are turned such that the driver perceives themselves following the line of the road the direction the car actually points (its heading) will not be the same as the heading of the driver; similarly if the road has a camber. But that doesn’t matter as it is perception we are controlling not physical values of the world.
BA: However, unless the disturbances overwhelm the system’s ability to compensate, the average heading will remain parallel to the road.
RY: However, in all cases, if the driver is controlling her perception of the position of the car then that will match her reference.
BA: No, not at all times. Disturbances will cause her car’s position to deviate from her reference position until she can compensate for their effect.
RY: Yes, of course. But she is constantly and continuously controlling her perception so that any deviation doesn’t build up.
Well, you did say “In all cases . . .her perception … will match her reference.â€?
BA: But what about a disturbance consisting of a constant crosswind? She will be able to compensate for the crosswind only by keeping the car’s steering angle turned against the wind.
RY: Yep, so the specific “steering angle” does not correspond to the direction in which the driver is going.
RY: With PCT all that matters is that the controller has the means to affect the perception. In this case the driver does this by turning the steering wheel which affects the perceived car position or direction. What happens between the steering wheel and the perception is the feedback path. That is, all the physical linkages within the car and all disturbances in the environment. They will all have effects on the perception.
RY: But PCT shows us two things about this,
RY: 1/ we do not need to know anything about those linkages, or about the disturbances within the feedback path.
2/ it would (generally) be impossible for us to know about all the effects on the perception within the feedback path.
RY: All that is relevant is that what we do affects our perception.
RY: I may be misunderstanding you, but you seem to be focussing on what is happening within the feedback path, and what are the different effects of those linkages. As far as PCT control is concerned they don’t matter, and are unknowable. It doesn’t matter if there are “misalignments”, or disturbances, PCT is able to overcome them without needing to know about them because what is being controlled is the perception not the physical values of the world.
I think we have been working at cross purposes. You want to jump right into the full situation, with a human driver at the wheel and disturbances at work. I have been analyzing how the proportional control system illustrated in the video works, for the purpose of demonstrating why the narrator brings steering angle into the discussion. Toward that end, I have been doing what scientists and engineers do when attempting to understand some physical system: start with the “ideal� case. You have me in the position of the scientists who says “a body in motion will stay in motion; a body at rest will stay at rest,� only to have the other guy immediately reply “that’s obviously wrong because of friction and other disturbances.�
The ideal case does not exist in nature, but nevertheless it often proves extremely useful in understanding how the system in question behaves. Once this understanding has been gained, one can move toward real systems by introducing various factors that may affect those real systems.
So, in the ideal case in which the steering is perfectly aligned and no disturbances are acting, the car will travel in a straight line so long as the steering angle is zero (straight ahead). Imperfections (e.g., slight misalignments, slightly different tire pressures, road irregularities, etc. sooner or later will cause the car to depart from straight ahead, but later in the discussion, when the control system is at work, these will be compensated for and when following a straight road, the car eventually will settle into a path that is parallel to the road except for temporary deviations caused by disturbances. Because of imperfections in alignment etc., this may be achieved by the controller maintaining some degree of deviation from the angle that would have produced straight-ahead driving in the ideal case.
What I have been emphasizing throughout is that in the ideal case, zero position error will produce a zero steering angle in the ideal case. Neither the observer nor the control system has to “know� this, it’s just a fact of the system’s design. For pure proportional control, O = ge, where O is the output, g is the gain of the output function, and e is the position error.
After the car has hit those rocks, the steering is misaligned and the car nor steers to the left when the position error is zero. How does the proportional control system deal with this? The car’s leftward drift produces a position error in the negative (leftward) direction. The error is positive, however, because the feedback is negative. Consequently the output function changes the steering angle toward the right, reducing the leftward drift. The leftward drift will cease when the net steering angle is again zero – the rightward turn of the wheels compensates for the leftwardd net angle of the misalignment.
At this point the car is still left of reference (lane center). The wheels are angled right to compensate for the leftward misalignment and the car is moving straight down the road. It will not move back to the reference position, because to do so would reduce the position error, and this would cause the steering angle to change so that the car would start turning left again. So the controller ends up keeping the car traveling down the lane but at a slight offset from reference, the size of which is inversely proportional to the loop gain.
So that’s the performance of the ideal proportional controller. The steering misalignment is acting like a crosswind that is pushing the car leftward and the controller is compensating by angling the wheels a bit more rightward than before the misalignment. The controller neither knows nor cares about the steering angle, but it must behave with respect to it, and when the misalignment occurs, this has consequences for the wheel-angle that the controller must produce if it is to keep the car moving parallel to lane center.
You seem to be claiming that the human controller would simply move her steering wheel appropriately to compensate for the misalignment and continue to track down the lane with no deviation from the reference (except for temporary deviations due to variable disturbances such as road imperfections and wind).
My analysis shows that the human controller would NOT behave this way if her control system is a pure proportional one. This immediately shows us that pure proportional control of lane position does not provide an adequate model of the human controller in this situation.
The video suggests one solution: add integral control. As the video demonstrates, the addition of an integral term in the output function allows the controller to “null out� the constant average positional error that is produced by the steering misalignment. Do human drivers function as PI controllers of lane position? I rather doubt it, for reasons I won’t elaborate here. But can we at least agree that the P and PI controllers will exhibit the behavior described in the video under the two scenarios (pre- and post-misalignment)?
Bruce
[Rick Marken 2018-09-02_10:21:24]
RM: A good description of the difference between the PID and PCT approaches to control is provided by my review of Jagacinski and Flach (2002) Control theory for humans: Quantitative approaches to modeling performance (NJ: Erlbaum):Â
https://www.mindreadings.com/BookReview.htm
RM: The Jagacinski and Flach book is an excellent introduction to the PID (or engineering) approach to control, as applied to human behavior. Some good introductions to the PCT approach to control are provided in the reference section of my review. Â
BestÂ
Rick
···
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery
[From Rupert Young (2018.09.09 12.45)]
(
I was referring to the cases you had listed, I didn’t say at “at all
times”.
Yes, I’d moved on, as I said, to what PCT systems are doing and that
the specific implementation is a distraction to the rationale of PCT
systems.
This is the problem; that leads to a control of output view of the
world.
Rupert
···
Bruce Abbott (2018.09.02.1125 EDT)]
RY: However, in all
cases, if the driver is controlling her perception of the
position of the car then that will match her reference.
BA: No, not at all times. Disturbances will cause her car’s position to deviate
from her reference position until she can compensate for
their effect.Â
RY: Yes, of course. Butshe is constantly and continuously controlling her perception
so that any deviation doesn’t build up.
Â
Well, youdid say “In all cases . . .her perception … will match her
reference.�
RY: I may be
misunderstanding you, but you seem to be focussing on what is
happening within the feedback path, and what are the different
effects of those linkages. As far as PCT control is concerned
they don’t matter, and are unknowable. It doesn’t matter if
there are “misalignments”, or disturbances, PCT is able to
overcome them without needing to know about them because what
is being controlled is the perception not the physical values
of the world.
I think we have been working atcross purposes. You want to jump right into the full
situation, with a human driver at the wheel and disturbances
at work.Â
I have been analyzing how theproportional control system illustrated in the video works,
for the purpose of demonstrating why the narrator brings
steering angle into the discussion. Toward that end, I have
been doing what scientists and engineers do when attempting
to understand some physical system: start with the “ideal�
case. You have me in the position of the scientists who
says “a body in motion will stay in motion; a body at rest
will stay at rest,� only to have the other guy immediately
reply “that’s obviously wrong because of friction and other
disturbances.�
The ideal case does not exist innature, but nevertheless it often proves extremely useful in
understanding how the system in question behaves. Once this
understanding has been gained, one can move toward real
systems by introducing various factors that may affect those
real systems.
[From Rupert Young (2018.09.09 12.50)]
(Rick Marken 2018-09-02_10:21:24]
Very interesting. Though there is no mention of PIDs, so I guess you
are talking about the way the approaches are conceptualised. What,
would you say, is the main message to take from this about the
difference in conceptualisation?
Rupert
···
RM: A gooddescription of the difference between the PID and PCT
approaches to control is provided by my review of Jagacinski and Flach (2002) * Control
theory for humans: Quantitative approaches to modeling
performance* ( NJ:
Erlbaum):
I have not been following this thread closely. I hope you will forgive me if this has already been established and the debate just concerns interpretation of the Youtube presentation.
Responding to Rupert’s original question, isn’t PID just a heuristic proposal for the output function of a canonical control loop?
https://en.wikipedia.org/wiki/PID_controller#Fundamental_operation
···
On Tue, Aug 14, 2018 at 8:40 AM Rupert Young csgnet@lists.illinois.edu wrote:
[From Rupert Young (2018.08.14 13.40)]
Here's a nice description of a PID controller,
Is there anything here, would you say, that would distinguish itfrom the PCT perspective of a controller, even if just
conceptually?
–
Regards,
Rupert
[From Bruce Abbott (2018.09.09.1240 EDT)]
···
From: Bruce Nevin (bnhpct@gmail.com via csgnet Mailing List) csgnet@lists.illinois.edu
Sent: Sunday, September 9, 2018 9:00 AM
To: CSG csgnet@lists.illinois.edu
Subject: Re: PID v PCT
I have not been following this thread closely. I hope you will forgive me if this has already been established and the debate just concerns interpretation of the Youtube presentation.
Responding to Rupert’s original question, isn’t PID just a heuristic proposal for the output function of a canonical control loop?
https://en.wikipedia.org/wiki/PID_controller#Fundamental_operation
Yes, it represents an output function of a canonical control loop – one in which there are proportional, integral, and derivative terrms:
Output = kperror + kdΔerror + ki*∑error,
Where kp, kd, and ki are the respective output gain terms.
The argument is about whether engineers view the PID control system as one that controls output (actions) or input (perceptions). Rupert’s view, if I understand him correctly, is that the equations presented in the video demonstrate a focus on controlling output, since they are used to compute values of steering angle, (the system’s actions, or means of control). But our canonical control systems all contain an output function in which similar computations occur, leading me to reject this interpretation. The purpose of the video is to demonstrate the effect of each of these terms on the ability of the car to track the center of the lane, so it is natural that these terms in the output function would take center stage.
A secondary argument arose as to whether a canonical PCT control system with proportional control would develop an offset from the reference position following an encounter with rocks that created a steering misalignment. I argued that it would, but Rupert and Martin Taylor disagreed (again, if I understood them correctly). Shortly I will be posting results of two Matlab car-steering simulations I developed, one using a PID controller and the other using a PCT-style two-level hierarchical control system.
These results should settle the question.
Bruce
On Tue, Aug 14, 2018 at 8:40 AM Rupert Young csgnet@lists.illinois.edu wrote:
[From Rupert Young (2018.08.14 13.40)]
Here’s a nice description of a PID controller, https://youtu.be/4Y7zG48uHRo
Is there anything here, would you say, that would distinguish it from the PCT perspective of a controller, even if just conceptually?
–
Regards,
Rupert
[Bruce Nevin 2018-09-09_19:50:19 ET]
Looks like an excellent summary to me–thanks!
···
On Sun, Sep 9, 2018 at 12:41 PM “Bruce Abbott” csgnet@lists.illinois.edu wrote:
[From Bruce Abbott (2018.09.09.1240 EDT)]
Â
From: Bruce Nevin (bnhpct@gmail.com via csgnet Mailing List) csgnet@lists.illinois.edu
Sent: Sunday, September 9, 2018 9:00 AM
To: CSG csgnet@lists.illinois.edu
Subject: Re: PID v PCT
Â
I have not been following this thread closely. I hope you will forgive me if this has already been established and the debate just concerns interpretation of the Youtube presentation.Â
Â
Responding to Rupert’s original question, isn’t PID just a heuristic proposal for the output function of a canonical control loop?
Â
https://en.wikipedia.org/wiki/PID_controller#Fundamental_operation
Â
Yes, it represents an output function of a canonical control loop – one in which there are proportioonal, integral, and derivative terms:
Â
Output = kperror + kdΔerror + ki*∑error,
Â
Where kp, kd, and ki are the respective output gain terms.
Â
The argument is about whether engineers view the PID control system as one that controls output (actions) or input (perceptions). Rupert’s view, if I understand him correctly, is that the equations presented in the video demonstrate a focus on controlling output, since they are used to compute values of steering angle, (the system’s actions, or means of control). But our canonical control systems all contain an output function in which similar computations occur, leading me to reject this interpretation. The purpose of the video is to demonstrate the effect of each of these terms on the ability of the car to track the center of the lane, so it is natural that these terms in the output function would take center stage.
Â
A secondary argument arose as to whether a canonical PCT control system with proportional control would develop an offset from the reference position following an encounter with rocks that created a steering misalignment. I argued that it would, but Rupert and Martin Taylor disagreed (again, if I understood them correctly). Shortly I will be posting results of two Matlab car-steering simulations I developed, one using a PID controller and the other using a PCT-style two-level hierarchical control system.
Â
These results should settle the question.
Â
Bruce
Â
Â
On Tue, Aug 14, 2018 at 8:40 AM Rupert Young csgnet@lists.illinois.edu wrote:
[From Rupert Young (2018.08.14 13.40)]
Here’s a nice description of a PID controller, https://youtu.be/4Y7zG48uHRo
Is there anything here, would you say, that would distinguish it from the PCT perspective of a controller, even if just conceptually?
–
Regards,
Rupert
Â
[Rick Marken 2018-09-09_17:54:32]Â
[From Rupert Young (2018.09.09 12.50)]
(Rick Marken 2018-09-02_10:21:24]
RM: A gooddescription of the difference between the PID and PCT
approaches to control is provided by my review of Jagacinski and Flach (2002) * Control
theory for humans: Quantitative approaches to modeling
performance* ( NJ:
Erlbaum):Â
RY: Very interesting. Though there is no mention of PIDs, so I guess youare talking about the way the approaches are conceptualised.
RM: I take PID to be synonymous with “control theory”. So both engineers and psychologists are applying PID models to the behavior of their systems. PCT is just a particular way psychologists apply the PID model to behavior. Most psychologists (like Jagacinski and Flach) apply PID models to the behavior of living systems the way engineers apply it to the behavior of artificial control systems. A few psychologists (PCT modelers) apply PID models to the behavior of living systems the way a psychologist should apply it to the behavior of living control systems. The difference between the engineering and PCT approaches to applying the PID model to behavior is a difference in how the PID model is mapped to the behavior of the system. The engineering approach to PID sees the reference signal as an input coming from outside the system that drives system output; the PCT approach sees the reference signal as an input coming from inside the system that drives system input.Â
RY: What,would you say, is the main message to take from this about the
difference in conceptualisation?
RM: The main message is that PID (control) systems control what they perceive; the behavior of a PID system is organized around control of perceptual variables. Engineers understand this implicitly when they design systems with sensors that can perceive the aspects of the environment they want the system to control. Roboticists understand this implicitly for the same reason. A few psychologists understand this because they have read Powers and understood his demos. Most psychologists don’t understand this for reasons given by Powers (1978) in the sections entitled “Input Blunder” and “Man-Machine Blunder”.Â
RM: For a scientific psychologist, it’s important to understand that input is controlled so that one doesn’t succumb to some version of the behavioral illusion in one’s research. For the roboticist, it’s important to understand that input is controlled because this will aim design efforts toward building systems that can perceive the complex results the robot is being designed to produce and away from building systems that produce the outputs that would produce these complex results only in a disturbance free environment.
BestÂ
Rick
···
Rupert
–
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery
[From Rupert Young (2018.09.10 9.30)]
(Bruce Abbott (2018.09.09.1240 EDT)]
No, I’m talking about the conceptualisation of the controllers not
the equations.
Regards,
Rupert
···
The argument is about whetherengineers view the PID control system as one that
controls output (actions) or input (perceptions).
Rupert’s view, if I understand him correctly, is that
the equations presented in the video demonstrate a focus
on controlling output, since they are used to compute
values of steering angle, (the system’s actions, or
means of control).