[From Rick Marken (2006.07.06.1040)]
Rick Marken (2006.07.05.1845)--
I think a quantitative analysis of all this could be done using the equations for a quasi-static analysis of control described in Powers' 1978 _Psychological Review_ paper. But they could be used only if we have a quantitative definition of efficiency. Maybe I'll give it a shot (using my own definition of efficiency) sometime before I leave for China. It is kind of an interesting question: does the efficiency of control (I would say in terms of how much energy it uses to produce a particular level of control) vary depending on the parameters of control?
OK, let me try this. I'll base this analysis on Bill's difference equation approach to determining the relationship between loop gain and speed in a control system. The relevant difference equation is the one that computes change in output per unit change in input. The basic equation is
(1) qo(t+1) = qo(t) + K[F(qi(t)-qi*)-qo(t)]
This is equation 15 in the Psych Review article. qo (t) and qo(t+1) is the output variable value (think of it as the horizontal position of a mouse) at time t and t+1 respectively. K is a slowing factor (between 0 and 1) indicating the proportion of the new output that should be add to the current output to produce the output at the next instant, t+1. The larger the value of K the faster the output changes over time. F is a linear multiplier that corresponds to the "system amplification function". The larger F, the more "sensitive" the system is to error, (qi(t)-qi*), so the more output the system produces per unit error. qi and qi* are the input variable and reference for the input variable, respectively. In order to solve the close loop version of this equation, we have to take into account the fact that qi(t) = Gqo(t) + Hqd, where G is a linear multiplier that corresponds to the feedback function and H is the function that relates the disturbance variable, qd, to the input variable, qi(t).
Making all the proper substitutions and simplifications I get the following equation for that describes a change in output during one time unit:
(2) qo(t+1)-qo(t) = K(FG-1) qo(t) + KF (Hqd - qi*)
Now here's where the subjective interpretation begins. I take the multiplier of first term on the right of the equation, K(FG-1), as a description of the effects of the output variable on itself. It represents all the factors around the loop that lead from the output at time t, qo(t), to the new output, expressed as a change in output, qo(t+1)-qo(t). The multiplier of second term on the right, KF, describes only the organism mediated effects of input, expressed as (Hqd - qi*), on output. So it represents only the system (organism) factors in the loop that from input to output.
I propose that the efficiency of a control loop be defined as the ratio of the all factors contributing to system output, F(FG-1), relative to only the system's contribution (input) to the production of this output, KF. That is, I proposed defining the efficiency of a control system (CE for Control Efficiency) as
(3) CE = K(FG-1)/KF
which simplifies to
(4) CE = G - 1/F
This definition of efficiency makes a lot of sense to me. It says that the efficiency of a control system increases with increases in the size of the feedback function, G, relating output to input. It also says that increases in system amplification reduce efficiency, assuming F is negative (as it must be if loop gain, the product of G and F, is to be negative). For example, if G = 100 and F = -1 then CE is 101. If G = 100 and F = -100 then CE is 100.01.
Increasing G basically acts as a lever that magnifies the effect you have on the controlled variable. So if an organism controls in an environment that magnifies the effect of its output on controlled inputs (independent of the organism's own efforts) then the organism can control very efficiently (in terms of CE), in the sense that it has to contribute relatively little of it's own amplification, F, in order to control with high gain (the product FG is system gain and the larger FG is, in a negative direction, the better control is). So it's the amplification of the feedback function, G, that is the main determinant of the efficiency of control.
Two systems, with the same FG, will control equally well. But they could be quite different in efficiency. Assume that for one system F = -10 and G = 1000 and that for another system F = -100 and G = 100. The loop gain of both systems, FG, is the same: -10,000. However, the CE for the first system is 1000.1 while that for the second is 100.01. The first system is about 10 times more efficient than the second because it is operating in an environment that gives it 10 times the purchase, in terms of it's effect on the control variable.
Another way to look at it is like this: for the first system, a relatively small proportion of the energy for loop gain comes from F, which is the amplification provided by the system itself; for the second system, much more of the energy for loop gain comes from F. I consider the energy from the system itself as the system "input" to control loop's "output", which is the result of all sources of energy that contribute to loop operation. So the great G, the greater the control loop output per unit energy input.
I think the definition of control efficiency (CE) in equation 4 could provide an interesting basis for analysis of the evolution of control systems. It seems plausible that an evolutionary advantage would go to creatures that evolve to perceive and control those perceptual variables for which the particular environment they are in provides a high G feedback function. So if there are, say, three similar perceptual variables that an organism could perceive and control, the "successful" organisms -- the one's who end up successfully reproducing -- will be the one's that are designed to perceive and control the one out of those three perceptual variable to which the environment they are in -- their niche -- provides the strongest feedback connection, G.
Best
Rick
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