Here are some thoughts about "modern control theory" drawn from a
reading of the back pages of _Modern Control Engineering_ [Ogata, K.
(1970), Prentiss-Hall], the most recent source in my library. I realize
that this book is out of date and that I do not follow the mathematical
methods in it, but one works with what one has at hand both mentally and
physically.
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The goal of "modern" control theory, as I understand it, is to devise a
driving signal (a vector) r, working through matrix computations to
produce a driving function, another vector, u(t). The driving function
u(t) is such that the variables comprising a plant will be brought to an
optimal state within some criterion of time or accuracy, or some other
global measure of quality of performance. The function u(t) may entail
multiple variables, even hundreds.
In order for the driving signals _r_ that result in u(t) to be
determined, the inverse of the plant function must be observable and
computable. If it is not, then some method of approximating the inverse
within some range must be found. Ogata suggests the application of small
test signals to the plant, which can be used to measure its static and
dynamic characteristics. Iterative procedures can then lead to closer
and closer approximations to the necessary driving function.
The basic concept of control in this field is therefore not built on the
principle of negative feedback, but on a principle of lineal causation.
Furthermore, the plant is not conceived of as a modular system in which
the overall control problem can be broken down into smaller problems,
but as a whole which is optimized in one huge calculation. Either the
entire plant is controlled, or control fails.
The function u(t) is derived from the requirement that the vector
describing the state of the plant approach some desired state, which in
turn is defined in terms of some optimality criterion. The optimality
criteria, being global, are not specifications for any particular states
of any plant variables, but describe some overall state of the plant
such as optimal energy usage or (I suppose) some weighted sum of errors.
These criteria are thought of as the "goals" of the control process.
They are specified not by the control system, but by someone or
something operating from outside the control system.
Notice that the idea of a reference signal is not central to control in
this way of thinking. The only thing corresponding to a reference signal
is the vector r which produces u(t), the driving function (via a
function which is the inverse of the plant function). Since this signal
that produces the driving function is derived from the global optimality
criteria, it is subordinate to them, becoming whatever is necessary to
achieve optimality. Thus the optimality criteria correspond to what we
call "intrinsic reference signals" in PCT. These criteria are given from
outside the control process; in the case of organisms, they are possibly
a product of evolution. The actual state of the optimizing system would
be derived from sensors detecting the state of the plant, passed through
computing functions that calculate the variables in terms of which
optimality is to be judged. The error signals that result (absolute or
squared) then are the basis for calculating changes in u(t). The only
variable that is controlled in a negative feedback way is the optimality
measure.
The language in which modern control theory is expressed is matrix
calculus (in discrete form). This language is suitable for handling
large complex systems in a compact way. My main problem with this
language is that it completely hides all the details of what is going on
in the system; one simply learns the rules of matrix manipulations, and
then applies them blindly, with no conception of what is actually
happening to individual variables in the system. For a person like me,
who wants to connect every variable and operation in a mathematical
representation to the physical system being represented, this is
intolerable: I get no sense of understanding how the system works. In
trying to follow Ogata's introductory materials on matrix manipulations,
I realized why it is that I have always dug in my heels and refused to
learn this language. I feel that I'm learning cookbook rules and losing
all contact with reality. This is probably a very peculiar attitude, but
perhaps that attitude is part of optimizing my own control systems to
compensate for mental deficiencies. There doesn't seem to be much I can
do about it.
The interesting thing about this bird's-eye view of modern control
theory is that it is not actually incompatible with PCT or HPCT. The
concept of global optimality criteria is quite similar to the idea of
intrinsic reference signals, and the idea that departures from
optimality lead to changes in the organization of control is also
similar. In my conception of reorganization, I do not speak of "optimal"
values of intrinsic variables, but only of reference values. Whether
those reference values are optimal in any sense is not important; all
that matters is that they are inherited. If the related variables are
maintained near their reference states, the system is functioning in a
way we presume to be sufficient for survival of the individual, for a
while at least. Whether this is "optimal" in any sense is irrelevant in
a behavioral theory.
The main difference between HPCT and modern control theory is in how the
control system is assumed to be organized internally, below the level of
the "master driving signal" r. From the standpoint of the matrix
approach, HPCT proposes a way of partitioning the calculations so that
intermediate variables can be identified with experience and with
specific operations in the nervous system and body. It also proposes
that the inverses which are needed are generated not open-loop but by
intermediate closed feedback loops; a feedback loop is an excellent and
simple way of creating the same effect as taking an inverse, but without
actually deriving any inverses.
In the HPCT model, the "plant" that is the external world is not
controlled in one huge chunk, but through layers of processes that work
from the simple to the complex.
The most proximal variables, those easiest to control, are put under
local feedback control at the first level, the level where muscle forces
and lengths are generated. These loops are independently stabilized,
regardless of events that are more remote from the organism. The effect
of these control systems is to create a relationship between driving
signals from higher systems and the resulting perceptual signals from
the sensors. The relationship is far simpler than it would be without
the local feedback loop; the apparent dynamics of limb movements, for
example, are reduced (nearly) to those of a simple proportional movement
with viscous drag. Without the local feedback, the higher systems would
be faced with a sensory response that lags, oscillates, and interacts
with other outputs at the same level. The local feedback makes otherwise
interactive subsystems essentially independent of each other and reduces
the order of the dynamical equations.
The first layer of control systems thus simplifies the apparent response
of the plant, the environment, to driving signals from higher in the
brain. At the same time, it automatically opposes the effects of
disturbances that act immediately on the force, velocity, and position
outputs of the first layer of systems. Such disturbances, therefore, do
not have to be handled by any higher systems unless they become
unusually large. So the world that is experienced by the higher systems
becomes both simpler and less subject to perturbation than it would be
without the first layer of feedback control.
A second layer of control further simplifies the world and handles more
kinds of disturbances (which enter through sensory signals that are not
involved in the first level of control). And so it goes, layer by layer,
each layer perceiving and controlling a world derived from the
simplified world of the level below it, further simplifying the world
presented to any higher systems and shielding the higher systems from
more kinds of disturbances. Of course to us who are built this way, the
higher perceptions seem more abstract, and because we can manipulate
them in many ways, more complex. But in fact it is the lower-level
signals that are truly complex, representing the world with a degree of
detail which would swamp the computing facilities of the higher levels
if presented directly to them.
Modern control theory, of course, says nothing about these intermediate
layers of feedback control. It doesn't matter to this theory if the
inverses required are calculated directly, or by the implicit methods of
negative feedback. If we had equations to represent the organization of
each system at each level in the HPCT model, the modern control theory
approach would be to combine them into a single enormous matrix, and
then to try to compute the output vector u(t) in one unimaginably
complex computation.
In principle such a computation might exist, but in practice it could
probably not be carried out by any material system. The hierarchical PCT
approach offers a way in which this calculation could in fact be done by
a brain. Moreover, it offers a way in which simpler organisms could do
part of the calculation (omitting higher levels), and still produce
control processes that resemble those of human beings at the lower
levels of organization. Even a cockroach's leg control systems employ
velocity and position feedback, quite similar to the same processes in
human limbs, although somewhat differently implemented.
Although I understand little of matrix mathematics, I have seen
suggestions that large problems in matrix manipulation can be greatly
simplified by suitable partitioning of the matrices, especially if the
matrices have favorable properties. In effect, the overall matrix is
broken down into a set of submatrices, each of which is much easier to
handle and demands much less computing power. What I am suggesting is
that HPCT offers a way of partitioning the overall matrix describing
control by a human brain, a way that turns an impossibly complex overall
problem into a collection of relatively simpler individual problems.
In addition, HPCT offers a way to identify various aspects of the whole
system with classes of subjective experience, with control tasks of
different kinds, with areas of brain function, and with characteristics
that differentiate complex organisms from simple ones. And HPCT also
offers a way of effectively obtaining the inverses of various parts of
the external plant in a way that does not require actually computing
those inverses.
I hope this constitutes progress.
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Best to all,
Bill P.