[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