[Hans Blom, 960826]
Earlier this year prof. Pieter Eykhoff said goodbye to this/his
university with a final oration. He taught control engineering for 32
years, and his book entitled "System Identification; Parameter and
State Estimation", first published in 1974 by Wiley, was somewhat of
a classic in its field in the 1970's and 80's. It was from him that I
learned the close connection between control systems and models.
In his oration, Pieter pointed to an important publication that
influenced the direction that he took with his subsequent research.
In 1970, Conant and Ashby published an article "Every good regulator
of a system must be a model of that system" (Int. J. Systems Sci., 1,
89-97). Pieter notes that, according to the Science Citation index,
this publication has hardly been noticed. Yet its importance has
become especially clear worldwide in the last ten years or so.
Eykhoff gives an example of the meaning of this provocative statement
which, he notes, must have been met with quite some anger by the
control engineers who, in the 70s, had after all developed all those
successful PID-controllers that were so popular in that period and
which did not contain a model at all -- or so one thought.
That those models are not always clearly visible to the designer
requires some explanation. A control system, according to Eykhoff, is
"a means through which a variable or a number of variables (of some
process) are forced to behave according to a prescribed norm". One
important example of this is "response improvement", where the output
signal (what PCT calls the perception) ought to be as much alike the
input signal (what PCT calls the reference) as is possible, particul-
arly -- in what is probably the most difficult case -- when the input
signal makes a (step) change from one level to another one. Without
feedback, the situation is
u (t) ----------- y (t)
----->| P (t) |----->
···
-----------
On a step change of u (t), the response y (t) depends on the charact-
eristics (step response) P of the process, and the norm -- that y (t)
resemble u (t) as much as possible -- is probably not fulfilled very
well. But something can be done about this by a compensator that is
put into series with P, as follows:
u (t) ----------- ----------- y (t)
----->| Q (t) |----->| P (t) |----->
----------- -----------
The best situation will exist when Q = P^-1, the inverse of P:
u (t) ----------- ----------- y (t)
----->|P^-1 (t) |----->| P (t) |----->
----------- -----------
In this case, y will accurately follow u. There are usually practical
problems, however. First, P may only be partially known; in that case
its inverse P^-1 will only be an approximation as well. Second, P may
have physical characteristics (such as a delay) which cannot be
inverted. In such cases, a _model_ M of P is used as a compensator,
and its inverse must be physically realizable:
u (t) ----------- ----------- y (t)
----->|M^-1 (t) |----->| P (t) |----->
----------- -----------
Note that a model must meet two requirements: a model M of a process
P must be a good approximation of P, and its inverse must be
physically (or computationally) realizable.
Now comes a trick that is well-known to electronics engineers: an
inverse can be physically realized by a feedback system around an
amplifier, denoted by inf, with an infinite gain and a comparator
(together these form an operational amplifier):
u (t) ----- ----------- x (t) ----------- y (t)
----->| |----->| inf |------>| P (t) |----->
+ ----- ----------- | -----------
^ - |
> ----------- |
---------| M (t) |<--
-----------
In formula: x (t) = inf * [u (t) - M * x (t)]
or, if inf is (or goes to) infinity:
x (t) = M^-1 * u (t)
so that
y (t) = M^-1 * P * u (t)
which is approximately y (t) = u (t), and this is what the norm
prescribes.
It is even possible (beware, another trick!) to use the process P
itself, rather than its model M, in the feedback loop, as follows:
u (t) ----- ----------- ----------- y (t)
----->| |----->| inf |----->| P (t) |----->
+ ----- ----------- | -----------
^ - |
> ----------- |
---------| P (t) |<--
-----------
because this schematic is identical with
u (t) ----- ----------- ----------- y (t)
----->| |----->| inf |----->| P (t) |----->
+ ----- ----------- ----------- |
^ - |
> >
-----------------------------------------
In this strange (?) case, the process P functions, additionally (!),
as its own model. No separate, explicit model is required! Isn't this
a nice unification between the PCT model and the explicit model
approach? Eykhoff notes that a similar reasoning can be given for
disturbance suppression.
Another quote: "The purpose of a controller can be defined
mathematically in a variety of ways. This leads to a variety
of control algorithms, all of which have in common, however, that a
good description of the process (read: a good process model) must be
available. These algorithms are collectively denoted by the term
"modern" control theory."
So Eykhoff's definition of "modern" control theory is that they
contain _explicit_, not implicit models. But the above makes clear
that, implicit or explicit, models play a central role in control
systems. The problem is that they are sometimes so implicit that they
are invisible -- unless you know what to look for.
Greetings,
Hans