[From Rick Marken (2004.01.03.1200)]
Rick Marken (2004.01.02.1515)--
Yes, it's easy to incorporate "internal" disturbances into my
spreadsheet hierarchy. But it's really not necessary to do this since
these internal disturbances already exist in the spreadsheet in the
form of the effects of systems at the same level on the perceptions
controlled by other system at that level.
Yes, I know (and knew) that, but for didactic purposes those effects
represent something different. They are (in theory) predictable given
the structure and weights in the hierarchy.
Yes. But they are not predictable by the system controlling the
perception that is disturbed by changes in these other perceptions. The
main cause of unpredictable changes in these disturbing perceptions is
changes in higher level goals. This change can be seen in the
spreadsheet hierarchy by changing the level three goals (from 1 to -1
or vice versa) which are in cells D3-I3 and seeing what happens to the
net disturbance to the perception controlled by level 2 system 1 (what
I called p12) in cell D8. Recall that p12 is defined in the spreadsheet
as
p12 = a1*p11+a2*p21...a6*p61
Since the output of the system controlling p12 only influences the
value of p11, the rest of the perceptual components of p12 are a
disturbance to p12:
d12 = a1*p21+a2*p21...a6*p61
so p12 can be written in the form of the following formula:
p12 = a1*p11+d12
This formula shows that by taking the appropriate weighted sum of p21,
p31...p61 (cells E12-I12 in the spreadsheet) we can see how the
disturbance to p12 (cell D8) changes with changes in references to the
higher level systems. In the version of the spreadsheet I have (and
that I believe is on the net) the appropriate weights for p21,
p31...p61 (E12-I12) are 1,1,-1,-1 1, respectively. If you compute this
weighted sun in some available cell, you will be seeing the value of
the "internal disturbance", d12, to p12. This value changes slightly
over time (due to variations in lower level perceptions that result
from external disturbances) but it stays pretty close to some average
value (the weighted average of the references for the level 1
perceptions).
Large changes in d11 are created by changing higher level references.
For example, when the level 3 references for the logical perceptions
controlled controlled by level 3 systems 2, 3 , and 4 in cells E3-G3
are set to 1,-1, 1, d12 is about 52. When the same level 3 references
are 1,1,1, d12 is 108 and when they are 1,1,-1 d12 is 350. By the way,
I can't seem to get the level 3 systems to stabilize when these level 3
references are -1,-1,-1, indicating that my definition of the level
three (logical) perceptions has led to a conflict that occurs only when
the references for these perceptions are set at certain values. Fixing
this up in a real control system would be a job for the reorganizing
system.
Anyway, the point is that changes in the references for higher level
perceptions (level 3 in this case) change, in unpredictable ways (from
the point of view of the lower level system -- a level 2 system in this
case) the disturbances to the perceptions controlled by lower level
systems. Dealing with such disturbances -- whether they are caused by
variations in the environment ("external" disturbance) or by changes in
higher level references in a control hierarchy (internal disturbances)
-- is what control systems _do_. I don't believe that there is any
reason to deal with external and internal disturbances differently.
Disturbances are disturbances and a properly designed control system --
one that controls properly, keeping its perception matching its
reference -- acts to protect the controlled variable from such
disturbances. Internal disturbances are no more predictable that
external disturbances, at least this is true from the point of view of
the control system itself. Control systems are organized to bring
perceptions to references and maintain them at the reference value,
protected from disturbances, whatever the source of those disturbances.
A control system that is not resisting disturbances to a controlled
perception is either not yet skilled at controlling that perception or
is in conflict with another control system that is trying to control
the same or a similar perception relative to a different reference.
I think it would be useful to have random external disturbances
introduced at different levels.
That is very easy to do using the spreadsheet. Just add one of the
randomly varying disturbance values in cells D24-I24 to one of the
level 2 perceptions. For example, the formula for the perception I've
been calling p12, which is cell D8, could be changed from:
=IF(D6="*",D7,MMULT(P1i,Pmlt1))
to
=IF(D6="*",D7,MMULT(P1i,Pmlt1)+F24)
The formula for p12 is then MMULT(P1i,Pmlt1)+F24 (the first part of the
IF just allows for p12 to be an imagined value, equal to the value of
the reference for that perception). The MMULT does a matrix
multiplication of the lower level perceptions (P1i, which is the vector
containing p11..p61) by the corresponding weights (Pmlt1, which the
vector containing a11..a61). The resulting perception now has an
externally generated disturbance added (F24). You will see that the
addition of this disturbance causes no problem for the system
controlling p12 (D8); the system keeps the perception under control
just as well as it did when the only disturbance was the "internal"
disturbance consisting of the all perceptions (except level 1
perception 1, p11) which are not influenced by the output of the system
controlling p12.
That would show that the hierarchic model is in fact robust
against wide-ranging disturbances (I mean wide-ranging through the
hierarchy), and I'm not aware that such a demonstration exists.
I agree. And I just described how to do the demonstration using the
spreadsheet. I agree that it is a useful demonstration of the
robustness of control in a control hierarchy.
Best regards
Rick
···
On Saturday, January 3, 2004, at 08:34 AM, Martin Taylor wrote:
---
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400