[From Bill Powers (951009.1655 MDT)]
Kent McClelland (951008,9) --
The puzzle about conflict that you've run into is a consequence of using
control systems with integrating output functions. Such a a control
system responds to error with a rate of change of output. When the error
drops to zero, the output does not drop to zero; it remains constant
(rate of change goes to zero). So when you artificially set the
reference signals of the two opposing control systems to match the value
of the controlled variable, you reduced both errors to zero, and left
both outputs constant at opposing values.
To see a relaxation of the outputs, one method is to use a leaky
integrator in both systems. This means using an output function like
o := o + (G*e - o)/s,
where o is output, G is the steady-state gain, e is error signal, and s
is a slowing factor. The time-constant in units of 1 iteration is 1/s.
To avoid computational oscillations you can't make s any smaller than
1/(1 + G), but you can make it larger than that, by enough to get a
reasonable time constant. When you do this you will see the outputs of
both systems decreasing after you set the difference reference signals
to the same value.
Perhaps a more realistic way to get the same effect is to impose a cost
of acting on both systems. The amount of output produces both some
amount of benefit and some amount of cost. To simplify the model, you
can imagine each of the opposing entities consisting of two control
systems which send their error signals into the same integrating output
function. The output produces some amount of benefit b to one subsystem
and some amount of cost c to the other subsystem. In one subsystem, b is
compared with a reference signal b*, and in the other c is compared with
a reference cost c*. The two error signals eb and ec are inputs to the
common output function, so that
o := o + g*(eb + ec).
The controlled variables are
b = kb*o and
c = kc*o
and the error signals are
eb := eb* - eb, and
ec := ec* - ec
This compositive system will act the same as the single system as long
as kb is greater than kc.
You might want to try making the cost system a one-way system, so that
if the cost is less than the reference cost, the error ec is clamped to
zero. The conflicting control system would also be one of these
composite systems. This gives you four reference signals to play with,
eb1*, eb2*, ec1*, and ec2*. Adjusting them should lead to some
interesting cases.
I believe that the cost factor will result in the output leaking down to
the value where ec is a little greater than ec* in each system. If you
set ec* to zero for the conflicting composite systems, you should get a
relaxation of the conflict when you set the benefit reference signals to
the same level. How far it will relax will depend on how much cost each
system considers negligible.
In the model you are using, there is no cost of acting, so there is no
reason for either of the opposing entities to reduce its output to zero
when the conflict is removed.
···
-----------------------------------------------------------------------
Best,
Bill P.