Adding the H function that converts a disturbance to the disturbance effect on qi is definitely a thing to explore. The solution for the qo-qo in an ideal negative feedback system is:
qo = G^-1 [ H (qd) ]
The interesting phenomenon is when the disturbance function is the same as the feedback function, G=H. The inverse of G cancels out the inner G, and the equation simplifies to just
qo = qd
If H and G are nonlinear, the nonlinearities cancel out, we get a linear relationship.
AM: You also wrote in the previous post (twice) that the organism function is relating qd and qo, that is not correct. The organism function is relating qi and qo.
RM: Actually, we are both correct.
Nice damage control. You were referring to some other organism function.
Also, correct for the qo varying as derivative of qd.
Let me get back to this:
RM: Maybe this is the source of our apparently irreconcilable differences. This is an invention of yours; it has nothing to do with Bill’s 1978 paper. What you are saying is that qo and qi are not related by a function because there are other causal paths ending in qi. In Bill’s analysis in the 1978 paper the only other causal path ending in qi besides the one from qo is the one from qd. So you are saying that because qi = h(qd) + g(qo), qo and qi are not related by the function g(). Nothing like that shows up in Bill’ analysis probably because it is not true; qd and qo have independent effects (via independent functions) on qi so the functional relationship between qo and qi – the feedback effect of qo on qi – can be treated – indeed, must be treated – independently from the effect of qd on qi.
I’m saying that because qi = h(qd) + g(qo), it is not correct to say “qi and qo are related by function G”, because it is misleading. It implies that you can write qi = G(qo), but that is not correct, because that is not what the plots are going to show.
If you make a plot for an ideal negative feedback system, qi is going to be just one constant value (emphasize - in IDEAL negative feedback systems, in real systems it is going to be some low correlation cloud). So, qi is really not related (correlated) to any other variable in the ideal negative feedback system, even though there are at least two causal paths ending in qi.
(edited the upper two paragraphs a few times, not yet sure it is too clear. Qi is related to qo by an integral, which is a function too…)
Which brings me to the answer to what happens if the type of the controlled variable is different - nothing much. For good control, qi is always going to be more stable (have less variance) than some disturbing quantity, because behavior will vary to oppose the disturbance and keep qi near the reference value. To the extent any real system is well approximated by an ideal negative feedback system, the relationships hold. For low gain control, or difficult disturbances, or very nonlinear input functions, many things are possible, like your demo area vs perimeter shows.
RM: It seems to me that discussing the behavioral illusion without discussing controlled variables is like discussing the bent stick illusion without discussing the differential refraction of light in air and water.
Not at all. Qi represents all possible controlled variables. All the same to math or simulation. Just trying to stay on topic.
RM: And I also wonder why you are fixated on the S-R behavioral illusion described in Powers (1978).
I think it is a very interesting phenomenon, not just as the possible illusion in behavior research, but as a general property of negative feedback systems. It was used in analog computers to build inverse functions, and it is still used in control engineering and in designing amplifier circuits, etc, etc. Wherever there are feedback systems, there is input-output determined by the function in the feedback path, and mostly independent of the amplifier gain in the forward path. It is important to understand it right, can be useful for research.
RM: It doesn’t seem relevant to your power law of movement research. It’s only relevant to research where an environmental variable (qd) is manipulated under controlled conditions to determine whether there is concomitant variation in a behavioral variable (qo). In the power law research both variables involved in the power law – curvature and speed of movement – are behavioral variables. So there is no controlled variable being disturbed by one of those variables and protected from that disturbance by the other.
That sounds like something I would say (almost). Highly suspicious to hear it from you.
Ok, I’ll bite. What brings you to that conclusion after your previous conviction that the power law is an example of the behavioral illusion?