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that this book is out of date and that I do not follow the mathematical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.