[Martin Taylor 2012.04.11.09.35]
[Rupert Young 2012.04.10.19.00]
Responding to both Martin and Erling.
I'd like to avoid getting into tacking or other degrees of freedom, as this is not actually about sailing. What I am interested in is when control is entirely dependant upon an environmental "function"; in this case the wind. For modelling purposes, and my simple brain, I'd like to keep things simple so I'll restate the situation, though let me know if it's too simple.
No, it's not too simple. But your specific example admits of a simple solution, which I offer below even though Bill has already posted something similar.
Think of a small boat with a very basic sail that could be raised or lowered. The goal of the sailor is go forward, and so she raises the sail when the wind is from behind and lowers it when the wind is from the front, and so, respectively moves forward or stays still. The sail then can be in one of two states, 1 (up) or 0, the wind, -1, 0, or +1 (from behind) and the movement of the boat -1, 0, or +1 (forwards). The overall reference, then is +1, for the boat movement.
It would seem that it is necessary for the system (the sailor) to perceive the state of the wind (though not control it) in order to "decide" what to do; that is, if the wind is from behind (+1) raise the sail. So considering comments made is it better to think of controlling the relationship between the perceived state of the boat movement and the perceived state of the wind? In this way the system would use the wind to propel the boat forwards.
I don't think it is necessary for the sailor to perceive the direction of the wind if she is controlling the direction of motion and can actually perceive the direction of motion directly (as opposed to deducing it from a perception of the wind direction). Suppose she keeps the sail raised when the motion is in the wrong direction. What happens? The error in the controlled variable is large and sustained. According to PCT, what is the result of that? Reorganization. In this simplified system, what could be reorganized but the sign of the sail up-down variable? So what happens if the sail stays up when the wind is ahead? The sailor's reorganization leads her now to drop the sail when the motion is backward.
The opposite case follows the same logic. Let us set aside the effect of the wind on the hull, and assume the boat is stationary if the sail is down. The controlled variable is the perceived motion of the boat. If the sail is down, the controlled variable is in error permanently, so reorganization would lead the sailor to rise the sail. If the wind is from astern, the error goes to zero and reorganization stops. If the wind is contrary, the state is as above, and she drops the sail.
This simple system oscillates if the wind is from ahead. But that's not wrong, since that's the best you can do if you can't observe the environmental state. Consider what you might do during a power cut to see if the power has come back on. From time to time you would switch on a light to see if it worked. How often you would do this depends on how critical you feel that it matters. The same with the sailor. If she feels that the wind usually doesn't change much over an hour, she might try raising the sail momentarily every half-hour or so.
However, the above doesn't get to why I said your question is not too simple.
So, is the wind an additional perceptual input to the system? If control of the relationship between wind and movement (so both equal +1) how would this be modelled? I'll have a go at a spreadsheet model. Ideas welcome.
You certainly could look at the relationship between wind and movement, but that couldn't be a controlled variable because it is a fact of the environment. If you perceive the sail to be down, then movement is zero. If the sail is up, movement is with the wind. The problem is that the wind direction has to be related to the reference value for movement. If the wind is from the north and you want to go south, you raise the sail, but if you want to go north you lower the sail and wait until the wind turns.
You could control a perception of the relation between sail setting and wind direction, as Bill P. suggests, but that is not controlling the perception of boat movement. Instead, you are imagining that particular relations between wind and sail result in particular boat movements. You have no idea whether other factors may be disturbing boat movement. The sail my be down and the boat going backward because of the current when the wind actually is behind you and you could make way if you raised the sail. If you are not controlling a perception of boat movement, but only are controlling a function of wind direction and sail setting, you wouldn't do anything about the problem. In the more general case, of which this is an exemplar, there are all sort of reason why the motion perception might be subject to disturbance.
Let's look at an HPCT structure to deal with this. At the top, you have a control unit that controls perceived boat motion. At the bottom, where, to mix a metaphor, the rubber hits the road, you have two sensors of the outer world, one of which is for boat motion, the other for wind direction. And you have one effector acting on the outer world, the sail up-down value. The sail state is available to be sensed, even though (in HPCT) the action to change its state is not. How can these all be connected to the controller of motion perception so that, if the boat is going slower than its reference value, the sail is up when the wind is behind and down when the wind is ahead? By itself, the relation between sail and wind is not a problem, since it doesn't involve the perception of motion. It becomes a problem when you are dealing with the control of motion perception.
Forget for the moment that the sailor is controlling a perception of boat motion, and consider the perception Bill described, the relation between sail and wind. The sail can have a value +1 or 0, and the wind can have a value +1, 0, or -1. The product Sail x Wind (SxW) can have a value +1, 0, or -1. A SxW value of +1 means that the sail is pushing the boat forward, and a value of -1 means the sail is pushing the boat backward. Now consider the output of the control unit that controls motion perception. If the boat is going too fast, you want SxW to be -1. If it is just right, you want SxW to be 0, and if it is going too slow you want SxW to be +1. These are the reference values that would be supplied by the output of the motion perception control unit.
If the wind is unfavourable (W = -1), you cannot achieve SxW = +1. The best you can do is SxW = 0, and you have to live with the ongoing error in the motion control unit -- or reorganize and buy a motor. If the wind is dead calm, SxW will be zero no matter what you do with the sail (the environmental feedback connection is cut). Finally, if W = +1, you cannot achieve SxW = -1. and again there is nothing you can do to slow the boat down unless you reorganize and introduce some other mode of action. That leaves you with the possibility of reducing error in the motion perception control unit only if the boat is going too fast and the wind is unfavourable or is going too slow and the wind is favourable.
Since you are using this situation as an example of a more general question of relating changes in environmental feedback possibility to changes in action, just redo the above analysis while allowing the "S" variable to go negative (a sail setting that would slow the boat when the wind is favourable or speed it when the wind is contrary).
The organization of this small hierarchy has a velocity control unit above a relationship control unit, which is unconventional but plausible. At some level, there must be an element that perceives the behaviour and the relevant aspect of the environment, and controls a relationship between them. I believe such a connection does not exist in standard HPCT, though it is quite consistent with Perceptual Control Theory.
The only PCT-correct alternative to this variation of HPCT that I have been able to make work is a more radical variation that requires some control unit to perceive the reference value for the boat motion and control a relationship between that and the environmental feedback possibiities.
I hope that Bill or someone can correct me on this.
Martin