[From Rick Marken (2015.02.08.1145)]
RM: Negative feedback exists when the product of all the multipliers of the loop variable are negative. Consider the simultaneous equatoins that define a basic control loop:
o = r - p (1)
p = k.o*o + k.d*d (2)
RM: In these equations the only variables in the loop are p and o. Both r and d are independent influences on the loop so the sign of these variables doesn't count in determining whether these equations define a positive or negative feedback loop. So the polarity (positive or negative) of the loop defined by equations 1 and 2 is determined by the product of the signs of the effect of p on o and that of o on p. From (1) the sign of the effect of p on o is minus and from (2) the sign of the effect of o on p is plus. So the product of the signs is minus, indicating that these equations define a negative feedback loop.
RM: This procedure can be used to determine whether the equations that define a mass spring system describe a positive or negative feedback loop. The equivalent of equations 1 and 2 above for a mass spring system considered as a control loop are:
o = s * (r - Pos) (3)
Pos = - o + Force (4)
RM: If r is 0, as assumed by equilibrium theorists, then the polarity of this loop is positive since Pos has a negative effect on o and o has a negative effect on Pos. Since a mass-spring system does not behave like a positive feedback system (with exponentially increasing output) the loop can only be made stable by eliminating r so that
o = s*Pos (3a)
which is basically Hooke's law. So it is incorrect to imagine that an equilibrium system has a reference specification for the variable that is returned to its resting state when the Force that causes a displacement from this state is removed.
RM: When we solve the simultaneous equations for a control loop (equations 1 & 2) for p (the controlled variable) we get:
p = k.o/(1+k.o)*r + k.d/(1+k.o)*d
which, assuming k.o, the output gain, is very large, simplifies to:
p = r + (1/k.o)*d (5)
RM: When we do the same thing for the simultaneous equations for an equilibrium system (equations 3a and 4) we get
Pos = Force / (1+s) (6)
RM: Since Pos is the equivalent of p and Force is the equivalent of d equation 6 can be written as
p = d/(1+s) (7)
RM: Comparing equation 5 to equation 7 reveals the important difference between an equilibrium and a control system. Both are formally negative feedback loops. But equation 5 shows that, in a control system, the value of the input variable, p, is determined mainly by the reference specification,r. Equation 7 shows that, in an equilibrium system, the "input" variable (the position of the mass or of the pendulum bob) is determined completely by the disturbing force. There is no disturbance resistance at all.
RM: This is a very interesting discovery, I think. It shows that a system can formally be a negative feedback system (not technically an open loop system) and still act exactly like an open loop system. I think this explains why equilibrium systems -- negative feedback systems that act exactly like open loop systems -- could fool people into thinking that they are similar to control systems. In fact, equilibrium systems are not anything like control systems; they don't resist disturbance at all and they certainly don't bring variables to reference states -- they have no reference specifications. They are open-loop causal system systems masquerading as control-like systems by being analyzable as negative feedback systems. Equilibrium systems are very much l like Lewis Carroll's Boojum; they look like a Snark and so if you're hunting Snark you can be fooled into thinking you have a Snark when it's really a Boojum. Of course, the difference between Boojums and equilibrium systems is that when you find one, thinking you've found something like a control system, you don't softly and suddenly vanish away, as you can see.
Best
Rick
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Richard S. Marken, Ph.D.
Author of <http://www.amazon.com/Doing-Research-Purpose-Experimental-Psychology/dp/0944337554/ref=sr_1_1?ie=UTF8&qid=1407342866&sr=8-1&keywords=doing+research+on+purpose>Doing Research on Purpose.
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