[From Bill Powers (950712.1730 MDT)]
Oded Maler (950712) --
Why *in principle* are Equilibrium and Control different? Consider
a high dimensional universe consisting of many physical variables
interconnected via complex dynamical laws (you agree that something
like this is underlying everything, don't you?) Now you pick two
such physical variables that happen to correspond to some neural
signals and you call them P (perception) and R (reference). The set
of points of this huge state-space that satisfy P=R are indeed the
bassins of attraction of the system, to which the system will flow
from other regions in the state-space.
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 equalled 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).
The only point is that although the projection of the trajectory on
the P-R plane might be simple (we can assume R to be constant for
simplicity) these two variables are not observable at all (and will
probably never be), and the projection of the trajectory on the
*rest* of the world (the other variables) might look very complex
and not reveal what is going behind the scene.
You're too pessimistic. In the tendon reflex part of the arm control
system, the perceptual signal P exists physically as neural signals
generated by stretching a tendon. The reference signal R is a physical
signal reaching the motor neuron along with the perceptual signal. The
motor neuron acts as a comparator, emitting an error signal proportional
to R - P (P is in fact an inhibitory signal). And the error signal
causes the muscle to contract, pulling on the tendon.
I am confident that we will eventually identify all the variables in the
abstract PCT model (probably much revised) with actual neural signals
and neural computations. Come back in 1000 years and you'll see.
···
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Best to all,
Bill P.