Equilibrium versus Control Systems

[From Bruce Abbott (2015.02.10.1755 EST)]

Here’s a nice tutorial on the difference between equilibrium and control systems, from Bill Powers himself.

Bill Powers (950712.1730 MDT) –

Consider Rick’s example of a mass hanging on a spring attached to the

ceiling. If you pull down on the mass, it will move downward, but a

counterforce will develop which resists the movement. The mass will come

to an equilibrium position where the restoring force due to the stretch

of the spring equals the downward force applied to the mass.

Now imagine that we add a sensor that can detect the position of the

mass without affecting it significantly. The sensor signal goes to an

amplifier with a reference input, and the error signal drives a motor

that pulls upward on the top end of the spring (where it was attached to

the ceiling in the first case). When you pull down on the mass, the

spring will stretch. Equilibrium will again occur where the downward

force is just equaled by the upward pull from the spring. Now, however,

the mass will move downward by a much smaller amount. The reason is that

the top of the spring moves upward, so most of the stretch comes from

the movement of the top of the spring.

In both cases we end up with an equilibrium between opposed forces. But

in the first case, the equilibrium is passive, while in the second it is

active.

In the first case, the resistance to the applied force is generated by

the stretch of the spring, which in turn is generated by the applied

force. When you remove the force, the mass is raised back to its

original position, and the energy required to do this is at most the

energy stored in the spring, put there by the work done on the system by

the applied force. So this system is energy-conserving; all energy used

to apply the disturbance is recovered while the disturbance is being

removed.

In the second case, the work done in stretching the spring comes mostly

from the motor. The energy required to do this comes from a source

independent of the downward pull. This source, the power supply, is

continually being drained to maintain the torque of the motor against

the spring tension, and it must continually be replenished from

elsewhere in the environment. When the load is removed, energy is used

to turn the motor to let the spring contract again; some energy is

recovered from the spring, but a large part of it is lost. This system

is not an energy-conserving system.

The first is a case of passive equilibrium, the second a case of active

equilibrium. A control system establishes an active, not a passive

equilibrium, by drawing on an external source of energy.

The main difference between a control system and a passive equilibrium

system is in the loop gain: the product of all proportionality factors

encountered in one trip around the closed loop of causation.

Pulling down on the mass with the top of the spring stationary creates a

restoring force by stretching the spring. From the spring constant, we

can compute how much stretch will be created per unit of applied force.

Likewise, from the stretch of the spring, we can compute how much

(reaction) force there will be per unit of stretch. The product of these

two factors, the loop gain, is -1. The negative sign shows that the

reaction force is opposed to the applied force.

When the top of the spring is actively moved, pulling down on the bottom

of the spring would, by itself, generate a certain amount of restoring

force that we can calculate from the spring constant. However, the

control system responds by stretching the spring from the other end,

doing work on it and making the spring appear to be far stiffer than is

actually is. So while a given force would be predicted to stretch the

spring by some amount, the actual stretch of the spring is many times

that amount. The loop gain can be far greater than -1; it can be -10,

-1000, or even larger. This large loop gain is achieved at the cost of

expending energy which is not recoverable from the system.

So those are the main differences between passive and active equilibrium

(a better way to characterize the difference).

BA: Thanks, Bill! A model of clarity, as usual. I sure miss you.

Bruce

[From Rick Marken (2015.02.10.1800]

···

Bruce Abbott (2015.02.10.1755 EST)–

BA: Here’s a nice tutorial on the difference between equilibrium and control systems, from Bill Powers himself.

Bill Powers (950712.1730 MDT) –

RM: The problem with invoking what a person wrote as gospel, even if the person is Bill Powers, is that what they wrote may be in error. Moreover, what is “published” in this discussion group is not peer reviewed or otherwise vetted before it’s published. So there are likely to be mistakes in any particular post, as there are in this one from Bill Powers. Here is the first error:

BP In the first case [of the mass spring system - RM], the resistance to the applied force is generated by the stretch of the spring, which in turn is generated by the applied force. When you remove the force, the mass is raised back to its original position…

RM: The error is the implication that it is the “restoring” force alone that brings the mass back to it’s original position (Bill says “raised” because this is a mass suspended vertically on a spring) . But as we saw in an earlier part of this discussion the restoring force alone will just cause the mass to oscillate forever, never to return permanently to its original position. When the “disturbance” force is removed after displacement, the mass will go back and stay back in its original position only if there is a damping force (which is provided by gravity in the case of the suspended mass).

RM: But here is a more crucial mistake:

BP: Pulling down on the mass with the top of the spring stationary creates a

restoring force by stretching the spring. From the spring constant, we

can compute how much stretch will be created per unit of applied force.

Likewise, from the stretch of the spring, we can compute how much

(reaction) force there will be per unit of stretch. The product of these

two factors, the loop gain, is -1. The negative sign shows that the

reaction force is opposed to the applied force.

RM: Teh error is that -1 is not the loop gain of the loop in which the position of the mass is the variable being stabilized. Indeed, -1 is not a loop gain at all. The value -1 is just the sign of Newton’s third law relationship between applied force, Fa, and reaction force, Fr: Fr = -Fa. As Bill said (not quite correctly) from Hooke’s law (not the spring constant) we can compute how much stretch, x, will be created per unit of applied force: x = (1/k)Fa. And we can compute the reaction force from stretch: Fr = -kx. Solving for Fr we get Fr = -1*Fa which can be written as Fa = -Fr, which is Newton’s third law. So -1 is the coefficient that defines Newton’s 3rd law relationship between Fa and Fr; it is not a loop gain.

RM: Bill’s mistake about -1 being the loop gain of an equilibrium system invalidates his conclusion, which is stated here:

BP: So while a given force would be predicted to stretch the

spring by some amount, the actual stretch of the spring is many times

that amount. The loop gain can be far greater than -1; it can be -10,

-1000, or even larger. This large loop gain is achieved at the cost of

expending energy which is not recoverable from the system.

RM: The conclusion is that the difference between an equilibrium system and a control system is the size of the loop gain. This conclusion is based on taking -1 as the equilibrium system’s loop gain and, as we’ve seen, it’s not. It turns out that the loop gain of the equilibrium system, taken as a negative feedback loop, can be quite high,. This can be shown to be the case for the mass spring system where the loop gain turns out to equal the spring constant, k. The equations for the mass spring “negative feedback loop” equations are:

Fr = k*Pos

Pos =

  • Fr + Fa - Fd

where Fd is the damping force. The loop gain is given by multiplying the coefficients of the loop variables, Pos and Fr, which gives -1*k. Since k can be a very big number the loop gain of this system can be far greater than -1; it can be -10 or -1000 or even larger. So loop gain is not what distinguishes equilibrium from control systems. What distinguishes them is that there is no reference signal inside an equilibrium system and, therefore, no disturbance resistance.

RM: Since I’m sure you will write this off as nonsense again, how about telling us what the study of equilibrium systems is going to tell us about the controlling done by living organisms, which is kind of what PCT is about.

Best

Rick


Richard S. Marken, Ph.D.
Author of Doing Research on Purpose.
Now available from Amazon or Barnes & Noble

[From Rick Marken (2015.02.10.1820)]

···

Rick Marken (2015.02.10.1800)–

RM: I should note that while Bill is making a mistake here by considering an equilibrium system to have a loop gain of -1, he is explaining how to do the test for the controlled variable to determine whether or not a system is controlling a variable at all, even with low gain. The first sentence says that we can use Hooke’s law to determine the predicted stretch, x, of the spring that would be produced by a given applied force, Fa. That is, physics tells us that if we apply force Fa the resulting stretch will be x = (1/k)*Fa. If this is a control system the actual stretch produced by Fa will actually be many times [smaller] than that amount (leaving out the word “smaller” is another careless mistake on Bill’s part; if Fa actually resulted in an observed x many times greater than the predicted x then the conclusion would be that the system involves positive feedback).

RM: So Bill is explaining here what I have been describing as the way to determine whether or not a mass-spring system resists disturbances. It does resist disturbances if the displacement of the mass (equivalent to stretch of the spring), x, produced by a disturbance (applied force), Fa, is smaller than would be predicted based on physical law: x = (1/k)*Fa.

Best

Rick

RM: Bill’s mistake about -1 being the loop gain of an equilibrium system invalidates his conclusion, which is stated here:

BP: So while a given force would be predicted to stretch the

spring by some amount, the actual stretch of the spring is many times

that amount. The loop gain can be far greater than -1; it can be -10,

-1000, or even larger. This large loop gain is achieved at the cost of

expending energy which is not recoverable from the system.

Richard S. Marken, Ph.D.
Author of Doing Research on Purpose.
Now available from Amazon or Barnes & Noble