# Tail Wagging the Dog (was Feedback: Positive and Negative)

[From Bruce Abbott (2015.02.08.1845 EST)]

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.oo + k.dd (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.

BA:Â Fisherman: â€œCaptain, I almost had him!Â That swordfish was one helluva fighter – in fact at one point my reel wwas spinning so fast I had to pour water on it to keep the line from catching fire!Â But then he seemed to tire and I was able to bring him up beside the boat and Â raise him right out of the water.Â Then, just as I had his body level with the transom, he gave a massive jerk that parted the line and he got away, the hook still hanging from his lower lip!â€?

BA:Â Captain: â€œWell, donâ€™t feel too bad: that was no ordinary swordfish, it was Old Crafty you had on the line there.Â Many have tried to bring him in and just as many have failed.Â Heâ€™s fought many a hard fight and learned every trick in the book.Â Heâ€™ll try to convince you that up is down if he thinks it will help him win the contest.Â In fact I heard tell of one occasion when he convinced the sportsman holding the pole that he might just as well give up because the very idea of reeling Old Crafty in violated the known laws of physics.Â On hearing this the poor fisherman just gave up and cut the line.â€?

BA:Â When you do a mathematical analysis of a well-understood system that every engineering student learns about in school and you come up with a â€œvery interesting discoveryâ€? that no one but you has ever noticed before, you probably should question the validity of your analysis.Â That is especially true when the conclusion reached is that a negative feedback system acts like an open-loop system despite its being a negative feedback system (it must be magic!).Â How is it that in the system equations for your control system p is computed by adding output to disturbance, whereas for the spring system p is computed by subtracting output from disturbance?Â All the pathways in the system diagram are the same in the two cases, so the system equations should also be the same, save for a difference in gain.

BA:Â You were on the right track in your previous post, to which I provided a minor correction and asked if you now agreed with my analysis.Â You agreed that the mass-spring-damper system is a negative feedback system but then asserted that it does not resist disturbances.Â I explained that you arrived at this conclusion by making the wrong comparison.Â What you are doing is comparing the amount of resistance to disturbance produced by the spring (a negative feedback system) to the amount of resistance to disturbance expected from a spring (a negative feedback system), and finding no difference, conclude that there is no resistance.Â I pointed out that the correct comparison is between the deviation in position produced by the system under a given magnitude of disturbance, to the deviation produced with the feedback removed (converting it to an open-loop system).Â That is exactly how one evaluates the resistance to disturbance of a control system and it is done exactly the same way for the equilibrium system.Â It seems to me that you have ignored this fact deliberately in order to avoid having to admit that equilibrium systems do in fact resist disturbances.Â This behavior might be appropriate if your objective is to appear to win a debate rather than to arrive at a correct analysis of the system and its behavior. Â I hope thatâ€™s not the case.Â If not then thereâ€™s another possibility I could entertain.

BA:Â From your actions over the course of this exchange I might conclude that you are so committed to the idea that equilibrium systems either are or act like open-loop systems that you can only reason from this preordained conclusion back to a set of arguments that would appear to support this belief.Â In other words, youâ€™ve got the tail wagging the dog.Â In science we may have our prior opinions but try to keep an open mind and go with whatever the evidence dictates, even if it violates strongly held beliefs.Â We donâ€™t just ignore solid evidence that doesnâ€™t fit our theory, at least not if we are acting as scientists.Â Finding so-called â€œfactsâ€? to support a prior belief and ignoring those that donâ€™t belongs more to the realm of religion than to the realm of science.Â Itâ€™s how creationists dismiss or reinterpret evidence that supports Darwinian evolution and an age for the Earth of many billions of years.Â Itâ€™s not the kind of thinking that should be on display here is this, a supposedly scientific forum.

BA:Â So please, make the changes to your simulation code that I suggested and watch what happens when force is applied to the spring.Â Then remove the feedback and see what happens.Â Let the evidence speak to you.

[From Bill Powers (931130.1545 MST)]

I would consider it control if the (negative) loop gain was

greater than 5 or so. All systems in which the perturbing event

supplies the energy that is then used to restore equilibrium have

loop gains of 1 or less.

[From Bill Powers (941013.0705 MDT)]

If we define a control system as any negative feedback system with a

loop power gain greater than one, we can easily distinguish control

systems from negative feedback systems in general. Words like

“stabilized” are qualitative terms; what makes the difference is how

well a variable is stabilized, a quantitative question. A loop power

gain of 1 is a natural dividing criterion that doesn’t depend on whether

you “like the connotations” of a definition.

Bruce

[From Rick Marken (2015.02.08.2100)]

···

Bruce Abbott (2015.02.08.1845 EST)–

Â

Rick Marken (2015.02.08.1145)]

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.

Â

BA:Â When you do a mathematical analysis of a well-understood system that every engineering student learns about in school and you come up with a âvery interesting discoveryâ? that no one but you has ever noticed before, you probably should question the validity of your analysis.Â

RM: Extremely good advice. You should give it to the equilibrium theorists who “discovered” that the systems that are the most common demonstrations of Newtonian physics – harmonic oscillators – are actually quasi- negative feedback control systems.Â

Â

BA: That is especially true when the conclusion reached is that a negative feedback system acts like an open-loop system despite its being a negative feedback system (it must be magic!).Â How is it that in the system equations for your control system p is computed by adding output to disturbance, whereas for the spring system p is computed by subtracting output from disturbance?Â

RM: As I mentioned in my earlier post, the sign of the disturbance makes no difference as to whether the system has negative or positive feedback because the disturbance is added outside the control loop.Â

Â

BA: All the pathways in the system diagram are the same in the two cases, so the system equations should also be the same, save for a difference in gain.

Â RM: The problem with your analysis of a mass-spring system was placing a reference signal (of 0) in the system. So the input-output equation implied by your diagram,Â o = s * (r - Pos) , is wrong, as I discovered from simulation. When you use the correct input-output equation, o = s*Pos you can use ether Pos = -o + Force or Pos = -o - Force for the feedback (output-input) equation and you get the same result (except for a change of sign) when you solve for Pos (the controlled or stabilized variable):

Pos = Force/(1+s) or Pos = -Force/(1+s)

In either case the position of mass, Pos,is completely determined by the disturbance, Force. There is no disturbance resistance in a mass spring system; none at all. Not even the tiniest little bit.Â

Â

BA:Â You were on the right track in your previous post, to which I provided a minor correction and asked if you now agreed with my analysis.Â You agreed that the mass-spring-damper system is a negative feedback system but then asserted that it does not resist disturbances.Â

RM: Whether or not I agreed with your analysis is irrelevant now. My last post proved that the mass-spring system is, indeed, a negative feedback system (because the produce of the signs of the functions around the loop is negative) and that it does not resist disturbances at all (as shown by the fact that Pos = Force/(1+s)). It’s also shown by the fact that if you apply a force to a mass, the mass is displaced exactly the amount predicted by Hooke’s law.Â

Â

BA: I explained that you arrived at this conclusion by making the wrong comparison.Â What you are doing is comparing the amount of resistance to disturbance produced by the spring (a negative feedback system) to the amount of resistance to disturbance expected from a spring (a negative feedback system), and finding no difference, conclude that there is no resistance.Â

RM: That is not at all how I arrived at my conclusion. My conclusion had nothing to do with the amount of “resistance” to the disturbance. It had to do with the fact that when you push on the mass with force F the mass is displaced by exactly the amount predicted by Hooke’s law: 1/s*F. The disturbance is completely effective, which implies, of course, that there is no resistance to the disturbance.

BA: I pointed out that the correct comparison is between the deviation in position produced by the system under a given magnitude of disturbance, to the deviation produced with the feedback removed (converting it to an open-loop system).Â

RM:Hooke’s law is the prediction of the deviation from the resting position that would result with the feedback removed. If the deviation is x, Hooke’s law predicts that x = 1/k*Force. And that’s what we observe.Â

Â

BA: That is exactly how one evaluates the resistance to disturbance of a control system and it is done exactly the same way for the equilibrium system.Â

RM: Yes, that is exactly how we evaluate resistance to disturbance of a control system. We predict the effect of a disturbance from Newton’s laws and then see if the effect of the disturbance is what is predicted. In the case of the mass-spring system we predict that the effect of a Force disturbance to the mass will be a displacement x = 1/s * Force. Then we apply a force. If x is smaller than predicted then we know that there is a control system involved; there is disturbance resistance. If it is exactly what we predicted (as in the case of the mass spring system) then there is no disturbance resistance.

Â

Â BA:Â So please, make the changes to your simulation code that I suggested and watch what happens when force is applied to the spring.Â Then remove the feedback and see what happens.Â Let the evidence speak to you.

RM: I did. All of these analyses are based on the results of running the simulation. The simulation let me to my “great discovery” which is simply that the great error in the equilibrium theory of stability is the apparently very small matter of assuming that the resting position of the system corresponds to a reference signal in a control system. I discover this because when I ran the mass spring in the form of a closed loop system like the one you diagrammed (and making your suggested corrections) I found that it works (in the sense of behaving like a mass-spring system) only when the reference signal (0) is eliminated. The code that works is:

Â p = Pos

Â OutputForce = s * p

Â Acc = (Force - d * Vel - OutputForce) / Mass

Â Pos = Pos + (Vel + 0.5 * Acc * dt) * dt

Â Vel = Vel + Acc * dt

This doesn’t work if you have OutputForce = s*(0-p) – where (0-p) is the “error” signal, reference signal of 0 - p. Once you put in a reference signal of 0 (or any value) the simulation fails no matter what you do to the rest of the code.

Â

[From Bill Powers (931130.1545 MST)]

Â

I would consider it control if the (negative) loop gain was

greater than 5 or so. All systems in which the perturbing event

supplies the energy that is then used to restore equilibrium have

loop gains of 1 or less.

Â

[From Bill Powers (941013.0705 MDT)]

Â

If we define a control system as any negative feedback system with a

loop power gain greater than one, we can easily distinguish control

systems from negative feedback systems in general. Words like

“stabilized” are qualitative terms; what makes the difference is how

well a variable is stabilized, a quantitative question. A loop power

gain of 1 is a natural dividing criterion that doesn’t depend on whether

you “like the connotations” of a definition.

RM: All that’s very interesting but what I have shown is that the disturbance to the mass in a mass spring system has no effect so, for all intents and purposes, the gain of this negative feedback system is 0.

Best regards

Rick

Richard S. Marken, Ph.D.
Author ofÂ Â Doing Research on Purpose
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