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Cybernetics in the 3rd Millennium (C3M) --- Volume 6 Number 1, Jan. 2007
Alan B. Scrivener --- http://www.well.com/~abs --- mailto:abs@well.com
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"Everything Has To Go Somewhere"
~ OR ~
"Eigenvectors and You"
[There was no November 2006 issue for a number of reasons, including
a 4-week "hold" on some Apple laptop power supplies, heavy business
travel, and the previously mentioned "other things I gotta' do."]
THE VOTES ARE IN
Reading maketh a full man
conference a ready man
and writing an exact man.
-- inscribed in the lobby of the Library of Congress
For those of you who are new subscribers, last time (vol. 5,
num. 5, see archives below) I polled my readers on which
topics they would like me to write about.
Hoo boy. The readers have spoken. I tabulated all of your
responses (thanks you very much) and summarized them in an
HTML file:
http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/votes.html
The number one choice is, much to my astonishment, "Everything
Has To Go Somewhere" ~ OR ~ "Eigenvectors and You." This is
problematic because when I came up with that glib title I didn't
know what Eigenvectors were. Sure, I USED to know, back in 1990,
when I took class in control engineering from UCLA extension.
But in the intervening 16 years the knowledge faded. I was
bluffing. I didn't think you all would pick that one. Well, have
no fear, I've jumped back into the subject, and with the help of
some textbooks and the Internet I have come to understand the
true meaning of Eigenvectors better than ever.
A VECTOR OF DESIRE
"This is some kind of plot, right?" [asks Slothrop.]
"EVERYTHING is some kind of plot, man," Bodine laughing.
"And yes but, the arrows are pointing all different ways,"
Solange illustrating with a dance of hands, red pointing
fingervectors. Which is Slothrop's first news, out loud,
that the Zone can sustain many other plots besides those
polarized upon himself . . . that these are the els and
buses of an enormous transit system here in the Racketenstadt
[Rocketcity], more tangled even than Boston's -- and that
by riding each branch the proper distance, knowing when
to transfer, keeping some state of minimum grace though
it might often look like he's headed the wrong way, this
network of all plots might yet carry him to freedom. He
understands that he should not be so paranoid of either
Bodine or Solange, but ride instead their underground
awhile, see where it takes him. . . .
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel) p. 603
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
First an aside: in researching this 'zine I wanted to use the
above quote, but despite having a homemade index of the novel
I couldn't locate the passage in the stream-of-consciousness text.
All I remembered was Slothrop and the "fingervectors," but
not whose fingers they were. I thought it was when Slothrop met
Saure Bummer, the German cat burglar, or Der Springer, the film
director turned black marketer. No dice. I called a few friends.
TS said it was near the beginning when Slothrop met the Argentine
anarchists who'd stolen the German U-boat. Nope. TB said it
was in the dream sequence near the end, with all the Spy vs. Spy
stuff. Couldn't spot it there either. Finally in desperation
I asked my speed-reading wife to do a brute-force search, and
she found it in the scene where Slothrop is in the baths with
the German prostitutes and their clients.
But along the way I did some Googling and made some interesting
peripheral discoveries. A Pynchon message board had some nice
concise definitions of Eigenvectors:
( http://www.hyperarts.com/pynchon/v/extra/eigenvalue.html )
A reviewer
( http://www.eye.net/eye/issue/issue_05.15.97/plus/books.html )
of the novel "Mason & Dixon" (1997) by Thomas Pynchon
( http://www.amazon.com/exec/obidos/ASIN/0312423209/hip-20 )
mentioned his use of vectors and other mathematical abstractions
as literary metaphors:
Pynchon approaches history with the eye of an engineer, on
the lookout for vectors, forces and gradients. But it is
not a mechanistic view. Like Gravity's Rainbow, Mason & Dixon
is rich with metaphors like this one, the response of one
of the narrator's cousins to his aside that astronomical
measurements are a form of "celestial trigonometry, by which
the telescope transports us out into the sky to the object
we wish to examine."
"A Vector of Desire," murmurs the boy.
Pynchon is drawn to metaphors like this, equating human
behavior or emotions to mathematical relationships, but in
a sense that draws out either their apparent inevitability,
or our frail hope that we might understand the human toll of
our actions as easily as we look up a telephone number.
Bearing in mind that the similarly diversion-prone Herman
Melville wrote a very big book that is only somewhat about
a big white whale, Mason & Dixon is an account of the lives
of two men whose work is a quest for measurement, precision
and observation, playing their parts in a larger search for
knowledge of the world around and the world within.
And of course a few weeks ago his latest was published,
"Against the Day" (2006)
( http://www.amazon.com/exec/obidos/ASIN/159420120X/hip-20 )
with a length of 1085 pages, and it concerns -- among other
things -- high-dimensional vectors and their significance.
( http://www.complete-review.com/reviews/popus/pynchon.htm )
I have written about these in several other issues, including
the two-parter on Wolfram,
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0102.txt )
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0201.txt )
the one on the 2002 SIGGRAPH Conference,
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0206.txt )
and the one on simulation.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0502.txt )
DOING IT THE HARD WAY
Lobster sauce though a necessary adjunct to turbot, is entirely
unwholesome. I never ask for it without reluctance. I never
take a second spoonful without a feeling of apprehension on the
subject of possible nightmare. This naturally brings me to the
subject of Mathematics,...
-- Lewis Carroll
quoted in "Introduction to Continuous and Digital Control
Systems" (book, 1968) by Roberto Saucedo and Earl E. Schiring
( http://www.amazon.com/exec/obidos/ASIN/B000H4H4WG/hip-20 )
I've told the story before of how I wanted to learn systems theory
in college and couldn't seem to find it anywhere, but I dropped
linear algebra, and later stopped being a math major, because I
didn't "get" what the point was of state spaces and ordinary
differential equations (ODEs). What nobody told me was THIS WAS
THE MATHEMATICAL LANGUAGE OF SYSTEMS THEORY! Duh! What a fiasco.
It took me about 20 years to correct this mistake.
To do it the traditional way, one would learn arithmetic and geometry
in grade school, algebra and analytical geometry plus some pre-calculus
in high school, then in college introductory calculus, calculus of
multiple variables, calculus of complex variables, linear algebra,
ordinary differential equations (ODEs) and the dynamical systems
theory, by which time you would need to be an upper division math
major to get into the classes.
The tragic thing about this structure is that there a lot of other
students who could benefit from the systems theory, if it wasn't so
hard to get to. For example, pre-med students are only required to
take introductory calculus (and usually a "bonehead" version of that!)
But as medical students it would benefit them greatly to have their
mental toolkits include mathematical models of systems -- in fact,
some medical schools even teach these models, not counting on the
universities to prepare the students.
But what I'm aiming at now is some practical advice for people who
have finished their formal education, and just want to learn systems
theory on their own.
NEATNESS COUNTS
If a scribe makes an error in the transcription of a royal edict,
he shall be [text unintelligible].
-- "The Code of Hammurabi"
translated by Doug Kenny, National Lampoon
I used to tutor other students in math, and one of the first things
I told about them was the importance of legibility. You have to
be able to read your notes. You have to be able to tell a scalar
from a vector from a matrix, and a function from its derivative,
especially when there are little dots or arrows or prime marks involved.
I encourage people to learn the Greek alphabet, especially the
lower-case letters,
( http://people.msoe.edu/~tritt/greek.html )
and be sure they can tell the lower case sigma, xi and zeta apart.
(Note that lower case sigma is sometimes drawn more simply than
in the above link, as in the "six sigma" logo.)
( http://decker.typepad.com/photos/uncategorized/sigma.jpg )
I also recommend learning calligraphy -- maybe take a class or get
a book, but definitely get a pen, like the "Calligraphic Marker
Style Pen, Medium Tip, Chisel Edge, Waterbase Black Ink" by Faber
Castell/Sanford Ink Company.
( http://www.amazon.com/exec/obidos/ASIN/B0006YZQEA/hip-20 )
Practice drawing the Greek and Latin letters, and other mathematical
symbols such as radical, del operator, integral, product sign and
approximately equals.
( http://www-306.ibm.com/software/globalization/gcgid/arithspc.jsp )
When I was first learning calculus with some friends in the 12th
grade, we had all read the science fiction novel "A Canticle for
Leibowitz" (1959) by Walter M. Miller,
( http://www.amazon.com/exec/obidos/ASIN/0060892994/hip-20 )
in which, in the far future, after a nuclear holocaust, a Catholic
monk named Francis faithfully copies a blueprint of an electronic
circuit diagram, thinking it is a mysterious religious document,
and then spends fifteen years making an illuminated manuscript of it.
( http://en.wikipedia.org/wiki/A_Canticle_for_Leibowitz )
This inspired us to -- on occasion -- don monks' robes and attempt
to solve integrals using our calligraphic pens, as if it was some
medieval religious ritual. Perhaps this is going a bit too far,
"beyond the pale" as they say in Dublin, but I do recommend that
if you study math you have a degree of reverence for the material.
Something else that has helped me get a handle on math has been
writing computer programs. Mathematicians are often ambiguous,
in the name of generality. Is N an integer or a real number?
Is X a scalar or a vector? Is S real or complex? Is K a constant
or a variable, or even a function? But in computer languages the
ambiguity must be removed before the program can run.
The notations in computer languages are simpler, too.
Symbols can't wander over the page like a division line,
radical, or giant parentheses. Each character occupies its
own little glyph, sort of a Golden Rectangle, in a pre-allocated
slot on the line, usually 80 characters wide. Some operators
get their own symbols, like:
+ addition
- subtraction
* multiplication
/ division
& and
| or
(to use examples from C) while others are expressed like
functions, such as:
sin() sine
cos() cosine
tan() tangent
sqrt() square root
exp() exponential
log() logarithm
The Greek letters go away, but variables can have names with
more than one letter, like lambda, and charactersPerLine,
so the symbol-space is actually much larger. Spaces tend to
be ignored, except inside names (where they are not allowed)
and quoted strings (where they are treated as literals),
and unlimited parentheses often save an otherwise difficult formula.
You you never really understand an algorithm or formula,
in my humble opinion, until you program it. That's when you
collide with all the hazards, where "the rubber meets the road."
For example, I've read many, many texts and web sites that say
you can convert from x,y form to r,theta with
r = sqrt(x^2 + y^2)
theta = tan^-1(y/x)
Where tan^-1 is the "inverse tangent" function. Ignoring the
problem when x = 0, I must point out that when you divide y by
x, the sign information of both is garbled, because:
1/ 1 = 1
1/-1 = -1
-1/ 1 = -1
-1/-1 = 1
Whatever you pass to tan^-1(), it won't know what quadrant the
number is in, and may be off by 180 degrees (Pi radians).
( http://classes.yale.edu/fractals/labs/AffTransf/AffTransfBackground/Angles.html )
What's needed is a function I call "untan" that takes both x and
y as arguments, so it can do the quadrant calculation itself:
untan(x,y) = 0 if x > 0 and y = 0,
untan(x,y) = tan^-1(y/x) if x > 0 and y > 0,
untan(x,y) = Pi/2 if x = 0 and y > 0, etc.
RULES AND INTEGRITY
Hamilton was looking for ways of extending complex numbers
(which can be viewed as points on a 2-dimensional plane)
to higher spatial dimensions. Hamilton could not do so for
3 dimensions: in fact later mathematicians showed that this
would be impossible. Eventually Hamilton tried 4 dimensions
and created quaternions. According to the story Hamilton told,
on October 16 Hamilton was out walking along the Royal Canal
in Dublin with his wife when the solution in the form of the
equation
i^2 = j^2 = k^2 = ijk = - 1
suddenly occurred to him; Hamilton then promptly carved this
equation into the side of the nearby Broom Bridge (which
Hamilton called Brougham Bridge.) Since 1989, the National
University of Ireland, Maynooth has organized a pilgrimage,
where mathematicians take a walk from Dunsink observatory to
the bridge where, unfortunately, no trace of the carving
remains, though a stone plaque does commemorate the discovery.
-- Wikipedia entry for William Rowan Hamilton
( http://en.wikipedia.org/wiki/William_Rowan_Hamilton )
I've mentioned this before, but it bears repeating. The typical
K-12 math curriculum tells students lies like:
"You can't subtract a larger number from a smaller,"
and then a year or two later says:
"Well, actually, you can."
The kids get irate, and not without good reason. Nobody focuses
on rules more than kids. We're messing with them when we pull
these fast ones. It would be far better to say:
"Here is a game we can play,"
and then next year say:
"Here is another game."
Nobody gets irate when you teach them how to play Rummy and
Crazy Eights with the same deck.
One way we get ourselves painted into a corner in our teaching
of mathematics is when we insist that these mental constructs are
DISCOVERED, not INVENTED. We seem to know better when it comes
to games -- nobody claims Crazy Eights was discovered.
But by insisting that mathematics is timeless -- almost like
a deity -- we obscure the fact that it is an arbitrary human
creation, subject only to some constraints, like a poem which
must rhyme. One of the few authors who gets this right is
Wolfram (in his discussion of the search for Extra-Terrestrial
intelligence of all things). He argues that if we send the
digits of Pi to the aliens, they won't understand them; the
digits are too idiosyncratic to our approach to math. He
advocates a search for all possible mathematical symbol systems,
and finding the common elements to all of them, then sending
THOSE to the aliens.
So if you are one of the kids who was lied to in math class
and never quite got over it, I'm here to reassure you that
this isn't really about TRUTH, it's about CONSEQUENCES OF
ASSUMPTIONS. And I invite you play a new game...
BADLY TAUGHT
"I-it's about the shape of the tunnels here, Master."
[In the Mittelwerk, where V2 rockets were assembled under
a mountain, safer from British bombers.]
"...I based the design on the double lightning-stroke,
the SS emblem..."
"But it's also a double integral sign, did you know that?"
"Ah. Yes, Summa, Summa, as Leibnitz said."
...but [the architect's] genius was to be fatally receptive to
imagery associated with the Rocket. In the static space of the
architect, he might've used a double integral now and then, early
in his career, to find volumes under surfaces whose equations
were known, -- masses, moments, centers of gravity. But it's
been years since he had to do anything that basic. Most of his
calculating these days is with marks and pfennigs, not functions
of idealistic r and theta, naive x and y. . . . But in the
dynamic space of the living Rocket, the double integral has a
different meaning. To integrate here is to operate on a rate
of change so that time falls away: change is stilled. . . .
"Meters per second" will integrate to meters. The moving
vehicle is frozen, in space, to become architecture, timeless.
It was never launched. It will never fall.
...the Rocket, on its own side of the flight, sensed
acceleration first. Men, tracking it, sensed position or
distance first. To get distance from acceleration, the rocket
had to integrate twice -- needed a moving coil, transformers,
electrolytic cell, bridge of diodes, one tetrode (an extra
grid to screen out capacative coupling inside the tube), an
elaborate dance of design precautions to get what human eyes
saw first of all -- the distance along the flight path.
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel)
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
To add insult to injury, after lying about the rules your teachers
mostly did a crummy job of playing the game with you. I used to
joke that in public school they inoculated you against knowledge,
so you wouldn't catch it later in life. But an even better
understanding of the educational system was provided by
James Herndon in "How To Survive In Your Native Land" (1971),
( http://www.amazon.com/exec/obidos/ASIN/0671230271/hip-20 )
He said the purpose of school was not to teach but to separate
the sheep from the goats. It doesn't matter if it's badly taught,
the smartest kids will learn it anyway. Higher education is
assumed to be in short supply, and only a few will earn their
way in. Of course, this system is ill-equipped to educate in the
Internet Age, when all levels of education are PLENTIFUL.
What I would propose is getting some visually programmed
systems simulation software, like Stella which is good for
learning. Its price ranges from $1,899.00 for a commercial
user down to $59.00 for a K-12 student, so it seems affordable,
especially if it can be used for years.
( http://www.iseesystems.com/softwares/Education/StellaSoftware.aspx )
Teach students to model simple systems visually, and then let them
play with the resulting simulations, adjusting parameters in real
time as the models run. They WILL learn systems theory, and I'll
bet you can do it with secondary school kids.
( http://maps.unomaha.edu/maher/GEOL2300/week11/geol2300Stella/introtoStella.html )
Let them learn the equations later, in high school.
ON FALLING ASLEEP WHILE READING "COMPLEX VARIABLES AND THE LAPLACE
TRANSFORM FOR ENGINEERS"
When they went to the movies he would fall asleep. He fell
asleep during Nibelungen. He missed Attila the Hun roaring
in from the east to wipe out the Burgundians. Franz loved
films but this is how he watched them, nodding in and out
of sleep. "You're the cause-and-effect man," she cried.
How did he connect together the fragments he saw when his
eyes were open?
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel)
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
Oh, I forgot to mention, I'm combining this topic with two
others: "Calculus Without Proofs In the Digital Age"
and "Visualizing Conformal Mapping With Java Applets"
which seem to flow together (at least in my mind) with
the Eigenvector stuff. The latter topic came about
because I decided I needed another dose of rigor in
my life, and so began trying to read (for the second time)
"Complex Variables and the Laplace Transform for Engineers"
(1961) by Wilbur R. LePage.
( http://www.amazon.com/exec/obidos/ASIN/0486639266/hip-20 )
It was tough sledding. I can tell you that LePage is NOT
an engineer himself. It seems written from a mathematician's
point of view. I would've preferred fewer proofs and more
example problems. Because of the difficulty I was having,
I frequently fell asleep while reading. My mind just switched
off. I kept at it though, a half page at a time, and managed
to finish the book, only partly comprehending what I'd read.
One Sunday afternoon I was napping thusly after reading
about conformal mapping, and how it is usually pictured,
and I dreamed a new way to visualize the mapping algorithms.
I dreamed it in detail and I dreamed the C code to create the
data. I even had some variable names chosen. I dreamed the
same dream several times repeatedly, in vivid detail. My
unconscious wanted me to get this.
Late in the afternoon I woke up, had some iced coffee, and
wrote the C code. It worked as I had dreamed.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/conformal_mappings.jpg )
(Originally, because I was in a hurry to see the results,
I produced data in a vector format for AVS software,
but I later modified it to also produce vector files in a
Wavefront format that could be parsed by a Java Applet, so
I could share the result. More on this later.)
Enjoy the pictures for now, and presently I will explain
what conformal mapping is, after some prerequisites.
SYSTEMS ON A CHECKERBOARD -- SECOND TRY
I liked the stuff on digital video, that made sense, but the
whole thing about "the set of all possible systems" went ...
[gestures of something flying over head].
-- a 'zine reader
I got feedback that my explanation of systems theory on a checkerboard
-- in C3M Vol. 5 Num. 2, Mar. 2006 "Even Better Than the Real Thing"
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0502.txt )
-- wasn't as clear as I'd hoped, so I'm going to try again.
For those of you with a background in Information Science, what
I'm describing is a FINITE STATE MACHINE.
( http://en.wikipedia.org/wiki/Finite_state_machine )
For the rest of you, what I'm describing is similar to the children's
game "Chutes and Ladders."
( http://www.amazon.com/exec/obidos/ASIN/B00000DMF6/hip-20 )
I think my description last time suffered mostly from a lack of
illustrations. So here it is, redone, with pictures. (I tried
to hand-draw these at first, but then resorted to computer-
generated images because in most cases it was faster. Note the
arrow are missing their heads; I'm sure you can figure out where
they go.)
A Very Simple Game
Have you ever seen the prank where you hand someone a card that
says "how to keep an idiot occupied for hours (see over)"
printed on both sides? Well this idiotic game is like that,
only it's played on a checkerboard. Each square is numbered 1
to 64.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/checkerboard.html )
On each square is a small card thats says something like
"go to square 18" or some other number from 1 to 64.
In each round of the game, there is a different set of
cards on the checkerboard.
You play by placing your marker (perhaps a miniature Empire
State Building) on one of the squares (called the 'current
state of the system'), and then following the instructions on
the cards one after another.
Imagine in one round every card says "go to square 1" and so
clearly you have one square that you always end up on, and then
you stay there. In systems theory is this is called an "attractor."
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/cb1.jpg )
Imagine if in another round the left half of the board
pointed to square 1, and the right half pointed to
the opposite corner, square 64. Now the state space is
divided into two "basins" each with its own attractor.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/cb2.jpg )
Or imagine if each square pointed to the one above or to the right
or both, until all jumps ended up on the top or right side.
The square in the lower left corner with so many jumps leading
away from its its vicinity is called a "repellor."
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/cb3.jpg )
(The purple lines going off the board up and to the right
are a mistake; please disregard.)
Or imagine that all squares in the interior point to an edge
square, and all the edge squares are joined in a chain that
goes around the perimeter clockwise (i.e., on the bottom row
each square points to the one to the left, meanwhile on the
left edge each square points to the one above it, and so on).
Now we have an "orbit" which in this case is also an attractor.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/cb4.jpg )
It is amazing the number of distinctions that can be drawn by studying
this idiotic little game.
This approach is largely the one in Ross Ashby's classic "An
Introduction to Cybernetics" (book 1956)
( http://www.amazon.com/exec/obidos/ASIN/0416683002/hip-20 )
which is back in print and also free on-line.
( http://pcp.lanl.gov/ASHBBOOK.html )
And . . . he continues the analogy while generalizing to
the continuum (infinitely many states) thereby deriving the whole
of cybernetics.
I have reproduced pages 22 & 23 from Ashby.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/Ashby23.jpg )
Kindly disregard the sentence fragment that starts in the middle with:
an ants' colony we might observe all the changes that follow the
placing of a piece of meat nearby
(intriguing though that fragment may be), and begin reading where it says:
Suppose, for definiteness, we have the transformation
U: | A B C D E
V D A E D D
In Ashby's notation, this defines a TRANSFORM named "U" that takes
an initial condition of A, B, C, D or E (the STATE of the system)
and modifies it as indicated. This differs from my chessboard
representation in that the states are not in a 2-dimensional grid
(8x8) but instead a 1-dimensional array, like half an egg carton,
or one row of the chessboard. But it doesn't matter; all the ideas
and conclusions you can derive are the same.
The only other thing you need know to understand the page is the
notion that applying a transform repeatedly (ITERATING it) is
represented by U^n, where n is the number of iterations.
(You don't have to do the problems; you can stop with the paragraph
that ends:
These matters obviously have some relation to what is meant by
"stability", to which we shall come in Chapter 5.
It is almost breathtaking how the core ideas of systems theory are
presented in these two pages.)
CALCULUS WITHOUT PROOFS IN THE DIGITAL AGE
A function consists of two sets, a domain-set and a range-set,
and a function-machine that follows these rules:
RULE 1. The function machine can only process things in
the domain set and only produce things in the range-set.
RULE 2. The function-machine is unambiguous: the same
input always produces the same output.
-- Swann & Johnson, 1975
"Professor E. McSquared's Original, Fantastic and
Highly Edifying Calculus Primer"
( http://www.amazon.com/exec/obidos/ASIN/0913232173/hip-20 )
In my searches for better ways to learn (what Marshall Thurber
called "superlearning") I happened upon a comic book called
"Professor E. McSquared's Original, Fantastic and Highly Edifying
Calculus Primer" -- I think it was in the Whole Earth Review.
It teaches standard first year college calculus (the calculus of
Newton with notation by Leibnitz) but in a witty and charming
way, with some funny characters. It follows the standard sequence
of sets, inequalities, limits, derivatives, and integrals, but
with few proofs, and it does a particularly good job of making
LIMITS easy to understand. Usually they are almost baroque in
their definition, but Prof. McSquared compares them to a
"guarantee" that you can find a delta for every epsilon.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/EMC0002.jpg )
There is also a clever and funny representation of FUNCTIONS,
as two robots wearing sneakers named Grover and Alfred, representing
g() and f(), with each of them having a chute for a number to go
in, and a trap door where another number comes out.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/EMC0001.jpg )
Later (if memory serves) this concept is extended to a TRANSFORM,
which is a bigger robot which takes in a function robot and puts out a
different function robot.
This comic is one of the best resources available for learning
calculus using the symbolic, analytic approach. But just this year
I realized you could put together a much simpler calculus curriculum
if you just dealt with the digital case. Without infinitesimals,
without classical analysis, without proofs, it becomes almost trivial.
For example, our old buddy Galileo, in the process of inventing
scientific method, measured falling bodies and concluded that,
absent air friction, on the Earth's surface, a body falls 16*t^2
feet in t seconds. (Read the expression as "16 times t squared.")
Today we could do this experiment with laser rangefinder aimed
straight down at a falling ball bearing and sampled once per second.
Rounding off all fractional parts we would probably get these
measurements in feet:
0
16
64
144
256
400
What we have here is a FUNCTION of time. Let's call it g (for
gravity, or maybe Grover the robot with sneakers). It's a
function of time t, so we could write it as:
g(t) = 16*t^2
IF WE KNEW THE FORMULA. Otherwise all we know is g(0) = 0,
g(1) = 16, g(2) = 64, and so on up to g(5) = 400 -- all the
data we have measured.
In the process of inventing the calculus, Newton and Leibnitz
arrived at the analytical conclusion that you can TRANSFORM
g(t) into another function, g'(t) (say "g prime of t") called
the DERIVATIVE of g, representing the RATE OF CHANGE of g,
which is also the SLOPE OF G. They formulated rules for
taking derivatives, which in this case transforms:
g(t) = 16*t^2
into:
g'(t) = 32*t
If you are representing the POSITION of an object in feet at time
t in seconds with g(t), then g'(t) is the VELOCITY of the object
at time t in feet per second.
( http://www.glenbrook.k12.il.us/gbssci/phys/CLass/1DKin/U1L3a.html )
Or, you can skip the analysis, and just go through the list
subtracting each number from the next. This yields:
16
48
80
112
144
Unfortunately it's the wrong answer, but the error is always 16.
If you add 16 to each number in the list you get:
32
64
96
128
160
which is the right answer. This can be OK because the problem
vanishes in the next derivative (as we shall see) and you can make
the error smaller by making the sample time smaller, say tenths
of seconds.
Next Newton and Leibnitz defined the SECOND DERIVATIVE, in this case
g''(t) = 32
In representing objects this is the ACCELERATION, or RATE OF CHANGE
OF RATE OF CHANGE.
In our approximate digital representation, we just take the above list and again subtract each number from the next, getting:
32
32
32
32
Aha! A free falling body has CONSTANT ACCELERATION.
We can also run the process the other way. If we start with the
second derivative list (of all 32s) and add each number into a
running total as we go, we get the first derivative again.
If we do it again we get the original list of positions. This
process is called finding the INTEGRAL, and it is the opposite
(or "inverse transform") of the derivative. It represents the area
under the curve, and can be used to find averages, among many other
uses.
( http://www.wpi.edu/Academics/Depts/Chemistry/Courses/General/figD-2.html )
So, what's wrong with the digital approach? Well, as we saw, there
are errors. We have no idea of the derivatives and integrals
outside the range in which we have samples. If we need more
detailed information (about intervals smaller than the step
size) we don't have it. But all of these problems are made
less severe if there are more samples of smaller sizes over a
larger range. And Moore's Law (computing power doubles every
18 months) keeps making this cheaper and easier.
What's nice about the digital approach is it's easy to understand.
Think of it as mathematical First Aid. As a Boy Scout I was
taught how to apply a band-aid and when to call a real doctor.
We should be teaching our students how to do quick-and-dirty
digital calculus, and when to call a real mathematician.
ODE TO ODES
A differential equation gives the rule by which the state
of the system determines the changes of state of the system,
which then determine its future evolution.
-- Alan Garfinkel, 1983
"A Mathematics for Physiology" in
"American Journal of Physiology"
( http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=6624944&dopt=Abstract )
( http://www.as.wm.edu/Faculty/DelNegro/cbm/GarfinkelAJP1983.pdf )
Once you have a grasp of the concept of a derivative, such as g'(t),
it becomes possible to describe system theory as envisioned by Newton
to solve the problem of computing planetary orbits. He invented what
we now call ordinary differential equations (ODEs), and he called
"fluxions." His notation allowed us to write "the rate of change
of t is zero" as:
g' = 0
(He used a dot over the x instead of a "prime" mark, but I don't
have that key on my keyboard.) This then is an ODE, and we can
solve it by guessing a function that meets the requirement, such as
g = k
where k is some constant, like 32. Note that Newton has compressed
the g(t) notation by taking out the (t). In many bold moves in math,
notation has been made more minimal. (The man was a genius, what
can I say?)
In general we will be given
g' = {some expression of g and/or t}
and we will SOLVE the ODE by finding (by guessing if necessary)
a g that fits for all t.
A more complex ODE would be:
g' = g
At first it seems like a mind-bender: the rate of change of g IS g.
What function is its own slope? Well, one answer is g = 0. If g
is always zero, it never changes, so its rate of change is zero.
But Mr. Genius "I'm Isaac Newton and You're Not" wants the complete
answer, an EXPONENTIAL function, t to the power of some base.
g = e^t
It turns out the base is an irrational number about 2.7, called e
or "Euler's Number."
( http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 )
What? Well, don't worry about that. Once again, let's throw
out analysis and take the digital approach.
Instead of looking at g' = {some expression of g and/or t} as a
guessing game (find the function g that fits for all t) think of
it as a set of instructions for deriving the values from any
INITIAL CONDITION. For example, if we have:
g' = 0
and know that g at time t = 0 is 13, then the solution is:
g = 13
The value of g is 13 for all t, and its rate of change is zero
everywhere. We generate the values of g, called INTEGRATING the
ODE, by adding zero to 13 over and over.
If we have:
g' = g
and we know g = 1 at time t = 0, then we get g at t = 1 by
adding the rate of change at time zero, 1, to the total, 1 to get
2. We are following the instructions, what Garfinkel called the
CHANGE RULES, to integrate the ODE. Next, to get g at time
t = 2, we take the value 2 at time t = 1, add the rate of change 2
at time t = 1, and get 4. Continuing we get the series:
1
2
4
8
16
32
... an exponential series! The base is 2 instead of e, but this
creates only an error of constant scale that can be made arbitrarily
small by increasing the step size.
This technique can be generalized to multiple equations of multiple
variables. You start with a VECTOR which is the initial condition,
and use the change rules to change each value at each time step.
For example, the system of equations:
a' = b
b' = -a
Analysis shows that any sinusoidal function for a, such as sine
or cosine (with angle measured in radians) will satisfy these ODEs.
( http://en.wikipedia.org/wiki/Trigonometric_functions#Definitions_via_differential_equations )
Or you can take the digital approach again and just integrate
the formulas: at each time step new a = old a + old b, and new b
= old b - old a. Starting with initial conditions of 0 and 1,
this gives the series:
0 1
1 1
2 0
2 -2
0 -4
-4 -4
-8 0
-8 8
0 16
16 16
32 0
32 -32
0 -64
-64 -64
-128 0
-128 128
0 256
256 256
512 0
Well, things are exploding here, in what engineers call "hunting"
or "overcorrection."
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/sin_approx1.jpg )
But, again, if you reduce the step size the problem becomes less
severe. With only ten times as many steps the equations begin to take
on pretty good approximations of sine waves.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/sin_approx2.jpg )
The textbook example of this type of system from control theory is
a MECHANICAL OSCILLATOR, such as a pendulum with a long arm relative
to its swing path, or a mass on a frictionless table connected to a
wall with a linear spring. Call the FORCE on the mass f. Newton
said force = mass times acceleration, or f=ma. Since mass in this
problem doesn't change, force is proportional to acceleration.
Acceleration is defined as the rate of change of velocity, v,
which is the rate of change of position, p. And the force due
to the spring is equal to the negative position -p (the farther
off center you are the more the force pushes you back) times a
spring constant k. Setting m and k to 1 for simplicity, we have:
f = a
p' = v
v' = a
f = -p
Simplifying we have:
p' = v
v' = -p
which is our system of ODEs for harmonic motion. Huzzah!
Once again we have seen systems that "do nothing, oscillate or
blow up." These are the types equations modeled by a tool like Stella.
( http://www.iseesystems.com/softwares/Education/StellaSoftware.aspx )
This is the language of systems theory. If you use Newton's
analysis it is very complex, and only a few of the systems of ODEs
are solvable. (But it is beautiful, elegant, and powerful when it
DOES work.) If you use the digital approach it is simple, easy
to compute, works for every set of equations, and can be made to
have minimal error with brute force, but it only gives you answer for
a given initial condition and time step. You don't get the CLOSED
FORM SOLUTION that defines g for all t.
Like I said, know when to call a mathematician.
THAT'S WHY THEY'RE CALLED COMPLEX
Consider the following subtraction problem, which I will put
up here: 342 minus 173. Now, remember how we used to do that,
3 from 2 is 9 carry the 1 and if you're under 35 or went to
a private school you say 7 from 3 is 6 but if you're over 35
and went to a public school you say 8 from 4 is 6, carry the 1,
and so you have 169.
-- Tom Lehrer, 1965
"New Math" (comedy song)
on the album "That Was the Year That Was"
( http://www.amazon.com/exec/obidos/ASIN/B000002KO7/hip-20 )
The other important mathematical tool for systems theory is what
they call COMPLEX ANALYSIS, which is math done with those weird
and wacky IMAGINARY NUMBERS.
Actually, it's done with good old REAL NUMBERS multiplied by
that ghostly supposed quantity, the square root of minus one,
and then added to other real numbers, to make numbers called
COMPLEX. Mathematicians often write them in the form:
z = a + b*i
But engineers tend to write:
s = sigma + j*omega
They use j instead of i because it is easier to read, especially if
you're writing lots and lots of equations (as engineers sometimes do).
They use sigma and omega to draw attention to the fact that
when you raise a complex number to the power of e (that pesky
Euler number) the REAL PART, sigma, acts like an exponent,
while the IMAGINARY PART, omega, acts like an angle, resulting
in sinusoidal variation at some frequency and amplitude.
This is because of the Euler Formula,
( http://en.wikipedia.org/wiki/Euler%27s_formula )
perhaps one of the most Baroque in mathematics, which says:
e^(j*omega) = cos(omega) + j*sin(omega)
And this really reveals the power of complex numbers, and why
they are so useful to engineers. They allow a compact method
for expressing harmonic systems as well as compounding systems,
and hybrids of the two, and do so in a way that is easy to do
computations on.
And as bizarre as these numbers may seem (and "you ain't seen
nothin yet"), all of their behaviors follow logically from the
simple assumption that:
j = sqrt(-1)
SO WHAT THE HECK IS CONFORMAL MAPPING?
In cartography, a conformal map projection is a map projection
that preserves the angles at all but a finite number of points.
The scale depends on location, but not on direction.
Examples include the Mercator projection, the stereographic
projection and the Lambert conformal conic projection.
-- Wikipedia entry on "conformal"
( http://en.wikipedia.org/wiki/Conformal )
One of the chapters in "Complex Variables and the Laplace Transform
for Engineers" (1961) by Wilbur R. LePage
( http://www.amazon.com/exec/obidos/ASIN/0486639266/hip-20 )
dealt with CONFORMAL MAPPING. This is a fairly of functions that
take as complex numbers as input as well as produce them as output.
In other words g(s) has real and imaginary parts that are each some
some function of sigma and omega, the real and imaginary parts of s.
One of the conceptual problems with such a COMPLEX FUNCTION
is that it is hard to draw a picture of it.
A simple function with real inputs and outputs can be drawn
with a line graph. For example, the falling body distance
function g(t) = t^2 is graphed as a familiar parabola.
( http://www.sparknotes.com/math/algebra1/quadratics/section1.html )
Even multiple functions of real values can be plotted together,
on a line graph with different line styles or colors to identify
each plot.
( http://www.teleologic.com/archives/fed-spending-graph.jpg )
But a complex function has too much information for a line graph.
Each possible sigma and omega input has to generate a sigma and
omega as output. The most commonly used technique to visualize
these critters is to use uniform grid as input and look at how
it is distorted.
( http://mathews.ecs.fullerton.edu/fofz/conformal/c10.htm )
( http://math.fullerton.edu/mathews/c2003/complexfunreciprocal/ComplexFunReciprocalMod/Images/ComplexFunReciprocalMod_gr_166.gif )
Another technique is to take a recognizable image and perform the
mapping on it.
( http://www.lactamme.polytechnique.fr/Mosaic/images/JFC.61.D/display.html )
This is a popular image processing technique, for creating surreal
distortions of photographs
( http://www.flickr.com/photos/sbprzd/sets/72157594172266668 )
as well as more abstract patterns.
( http://www.brainjam.ca/fractals.html )
VISUALIZING CONFORMAL MAPPING WITH JAVA APPLETS
"We didn't know we couldn't do it."
-- my father, 12/16/2006
telling about the early days of
Pacific Southwest Airlines (PSA)
While reading about conformal mapping, I began to think about
a simple way to visualize complex functions. I've done a lot
of work with interactive 3D graphics, so I find it easy to think
in 3D. Even old-fashioned 3D vector graphics, now called wire-frame,
can be remarkably rich in subtlety. And they are computationally
cheap, making them real-time on the slowest of modern chips.
So I came up with what seemed like an obvious idea: visualizing
the mapping as connecting one chessboard with another using straight
lines. I imagine it as being like sticks of uncooked spaghetti,
each connecting a sigma and omega on the input board to another
pair on the output board. Simple but potentially intricate.
Then, as I mentioned, I fell asleep and dreamed I was writing C
code to create the data files.
I initially created data in an AVS vector format to look at the
data on a local computer, but it was always my intention to
create data for a Java applet that users could load from the web.
Well, I finished the conversion and posted the result.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/Java/conformal.html )
Assuming the applet works in your browser, you can grab each
3D wire-frame with the left mouse button and rotate it, to
view it from any angle. This should give a pretty good idea of
the 3D structure of the set of lines in each case. I have provided
a small menagerie of mappings.
You will see that the IDENTITY function:
g(s) = s
has each line go straight across, staying horizontal, because
each s maps to s.
The TRANSLATE function:
g(s) = s + c
where c is a complex constant, has each line at the same angle
and orientation to the chessboards, staying parallel.
The SCALE function:
g(s) = s*r
where r is a real value, has the lines spreading out, diverging as they move away from the source chessboard.
The ROTATE function:
g(s) = s*j
which uses the fact that multiplying by j rotates a complex point
by 90 degrees counter clockwise, seems to create a curving spiral;
it's hard to remember that each line is straight.
The SQUARE function:
g(s) = s^2
is interesting in that it visually shows the fact that both
s and -s, when squared, yield the same number s^2. Each destination
point has two lines leading to it, and the shape is symmetric about
the j axis.
The INVERSE function:
g(s) = 1/s
(where s is not the complex origin, 0 + 0*j, usually just called 0)
is very intricate. If my grid wasn't sampled at intervals of 0.5,
but instead with a smaller sample size, the lines would diverge
even more.
I don't have proof yet but I'm convinced this technique can
contribute to an understanding of complex functions and mappings.
DON'T VOID THE WARRANTY
Black holes are where God divides by zero.
-- Steve Wright
I remember while reading LePage I got frustrated with something
that I find in a lot of math books. I had a hard time seeing what
the point was. Not of the whole book, but of each individual
section. I couldn't tell hypothesis from conclusion, proof from
example, and goals from means to a goal (e.g., theorems from lemmas).
it seemed like one long narrative of notations and consequences
with very little connection to the problem-solving that an
engineer would need to do.
The author did spend a lot of time on the convergence of transforms
such as the Laplace Transform, that is, under what conditions the
equations are useful, and when they "blow up" and produce useless
infinities. I recognized that this was important. It's like the
"warranty" for a mathematical technique: regions of non-convergence
"void the warranty."
This got me to thinking about how I would like to be taught math.
I realized I want a vast, hyperlinked body of knowledge, modeled
after the "Encyclopedia Galactica"
( http://en.wikipedia.org/wiki/Encyclopedia_Galactica )
in Isaac Asimov's "Foundation" trilogy,
( http://www.amazon.com/exec/obidos/ASIN/0739444050/hip-20 )
only focused on math, an "Encyclopedia Mathematica." The I
remembered what Douglas Adams wrote in "The Hitch Hiker's Guide
to the Galaxy" (book, 1979)
( http://www.amazon.com/exec/obidos/ASIN/0517226952/hip-20 )
( http://www.mindgazer.org/dontpanic/thehitch.htm )
about the "Encyclopedia Galactica" and its fate:
In many of the more relaxed civilizations on the Outer Eastern
Rim of the Galaxy, the Hitch Hiker's Guide has already supplanted
the great Encyclopedia Galactica as the standard repository of
all knowledge and wisdom, for though it has many omissions and
contains much that is apocryphal, or at least wildly inaccurate,
it scores over the older, more pedestrian work in two important
respects. First, it is slightly cheaper; and secondly it has the
words DON'T PANIC inscribed in large friendly letters on its cover.
So perhaps a more casual, less imposing name would be more inviting,
like "The Websurfer's Guide to Mathematics."
I see each page describing a tool, like the Quadratic Equation
( http://en.wikipedia.org/wiki/Quadratic )
which solves:
a*x^2 + b*x + c = 0
as:
x = (-b +/- sqrt(b^2 - 4*a*c))/2*a
First and foremost would be the WARRANTY for the tool. This one's
s warranty would say that a, b and c must be real numbers, and
a cannot be zero, because then when we divide by 2*a it's division
by zero.
Also, each tool should come with PREREQUISITES, and a QUIZ,
so you can tell if you are ready to understand it, and know
where to go first if you're not ready.
I would want each tool to come with a proof, ON A SEPARATE PAGE,
so readers could be free to ignore it. But if they want to dig
in, each step in the proof should hyperlink to the RULE that
justifies it.
And each tool needs EXAMPLES WITH NUMBERS. I remember reading
texts that cover linear programming, such as "Introduction to
Operations Research" (1967-2001) by Hillier & Lieberman,
( http://www.amazon.com/exec/obidos/ASIN/0071181636/hip-20 )
and encountering typical production problems something like this:
We are producing three products, p1, p2, and p3, and they
sell for $2, $3, and $8 respectively. Our goal is to
maximize our revenues. Each product has a per unit
production cost of $1, $2, and $5, respectively, and we
have a budget of $300. We have demand for the products
that requires that we produce a combination of at least
50 units of p1 and p2. Also we have exactly 400 hours of
available production time and each unit requires 2, 4, and
5 hours of production time, respectively. (The numbers p1,
p2 and p3 must all be greater than or equal to zero.)
What mix of products will yield the highest profit?
The texts would then show how to set the problem up in a commercial
linear programming solver program (of which there are several,
including the Microsoft Excel spreadsheet program!) and let
it crank out the solution. But THEY DIDN"T GIVE THE ANSWER!
(By the way, in this case it is p1 = 100, p2 = 0, and p3 = 40,
which has a value of $520.)
For the example of the Quadratic Equation, a simple
numerical example would be for a = 1, b = 2 and c=1:
x^2 + 2*x + 1 = 0
therefore:
x = (-2 +/- sqrt(2^2 - 4*1*1))/2*1 = (-2 +/ 0)/2 = -2/2 = -1
Plugging -1 back into the equation confirms the answer.
Some history of the tool, again on a separate link, would be
nice.
Lastly I would like to see some type of interactive visualization
of each tool. The Java Applets I wrote to visualize conformal
mapping would be an example of this. For the Quadratic Equation
I did a quick prototype that plots in a 3D space formed by a,
b and c, and draws a sphere with radius proportional to x. Green
spheres are the first real root, red are the second real root, and
cyan and yellow are pairs of complex roots (which always occur in
conjugate pairs) where sphere radius is proportional to the complex
number's radius, r, or distance from the origin.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/quadratic_roots )
To complete this I would need to make it interactive. The history
of the evolution of our big brains, as well as my own experience
teaching visualization, indicate that hand-eye correlation are
important to learning.
THE LINEAR CASE
A gangster who loved to bet at the racetrack kidnapped a
chemist, a mathematician, and a physicist to force them to
find ways for him to find money at the track. He gave them
all a month and he threatened to kill all three if they didn't
come up with anything useful, then he locked them up in labs.
The month expired, and the gangster first went to the chemist
and said, "So, what do ya got for me?"
The chemist said, "I've created this new variation on
amphetamines that there's no test for because it's new.
Give this to the horse before the race and it'll make him
run faster, and at least for a while it'll be undetectable."
The gangster said, "Great, go stand over there and wait."
Then the gangster went to the mathematician and said, "So,
what do ya got for me?"
The mathematician said, "I've found some flaws in the way
tracks calculate the betting odds. If you follow these
instructions, it'll increase your chances of walking away
a winner."
The gangster said "Great, go stand over there and wait."
At last the gangster went to the physicist and said, "So,
what do ya got for me?"
And the physicist said, "Consider a spherical horse in
simple harmonic motion..."
-- "Lets post nerdy jokes"
( http://www.world4ch.org/read.php/sci/1137494383/l40 )
Having slogged through LePage I picked up another book lying
around the house "Introduction to Continuous and Digital Control
Systems" (book, 1968) by Roberto Saucedo and Earl E. Schiring,
( http://www.amazon.com/exec/obidos/ASIN/B000H4H4WG/hip-20 )
which was given to me years ago by a friend I have since lost track
of, last seen working at Electronic Arts in the 1980s.
This book seemed like it WAS written by engineer, but I didn't have
all the prerequisites. I dipped into it anyway, and learned a few
more things.
Control theory can be divided into analysis and design, or in
other words, prediction and control. On the control side, the
distinction is drawn: if you want to force a variable to be
constant, that is a control system; if you want it to follow
an arbitrary signal, that is a servomechanism. (If what you're
after isn't a point attractor, you must be messing with robots?!)
In analysis, there are the classical frequency methods (using transfer
functions) and the modern state space methods.
A textbook example of classical analysis in this text is a DRIVEN
MECHANICAL OSCILLATOR WITH SPRING AND DAMPER.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/ICDCS0003.jpg )
( http://en.wikipedia.org/wiki/Harmonic_oscillator )
One instance of this would be a mass and spring as above, with a motor
as the driving force and a dashpot
( http://en.wikipedia.org/wiki/Dashpot )
as the damping force.
Another instance is an electrical circuit with a resistor (R),
and inductor (L), and a capacitor (C), called an RLC circuit.
( http://en.wikipedia.org/wiki/RLC_circuit )
These types of systems are popular with mathematicians because
they are LINEAR, and therefore SOLVABLE.
( http://en.wikipedia.org/wiki/Linear_system )
As Wikipedia explains, a classical linear system is one in which the
... given two valid inputs, x1(t) and x2(t), as well as
their respective outputs y1(t) = H(x1(t)) and y2(t) = H(x2(t)),
then a linear system must satisfy
alpha*y1(t) + beta*y2(t) = H(alpha*x1(t) + beta*x2(t))
for any scalar values of alpha and beta.
A whole toolbox exists to analyze these systems, including the
Laplace Transform for continuous systems,
( http://en.wikipedia.org/wiki/Laplace_transform )
and the Laurent Transform, also called the z-Transform, for
analyzing systems producing pulses at discrete times (or systems
sampled at discrete times).
( http://en.wikipedia.org/wiki/Laurent_transform )
The transforms are part of a family of methods based on a
TRANSFER FUNCTION relating the output to input. From my
notes in that control theory class at UCLA:
1) TF = Transfer Function = Input/Output ("black box") models
SS = State Space = parametric or "grey box" models
2) TF assumes system is relaxed, only relates I/O info; it does
not provide any info about system's internal interactions
3) TF may however be easier to obtain
4) TF is for single-input-single-output (SISO) & time
invariant systems; SS models can be multi-input-
multi-output (MIO) and time-varying
5) SS more suitable for digital simulation
A related concept is the LINEAR SYSTEM OF DIFFERENTIAL EQUATIONS.
( http://www.egwald.com/linearalgebra/lineardifferentialequations.php )
The more recent state space tools (mostly 1960s vintage) are based
on taking an arbitrary linear system of ordinary differential
equations (ODEs), such as:
w' = a1*w + a2*x + a3*y + z*a4
x' = b1*w + b2*x + b3*y + z*b4
y' = c1*w + c2*x + c3*y + z*c4
z' = d1*w + d2*x + d3*y + z*d4
(where x, y, z and w are variables while the values of a1 through
d4 are constant) and expressing it as a MATRIX, in this case
a 4 by 4 matrix:
| a1 a2 a3 a4 |
| b1 b2 b3 b4 |
| c1 c2 c3 c4 |
| d1 d2 d3 d4 |
and then giving that matrix a name, like M. The important thing
is that, in this context, M means the same thing as the 4 ODEs.
(Again, the genius is in removing something.)
The system is called LINEAR because each variable's rate of change
is a sum of linear combinations of the system variables.
A list of values for all of the variables, in this case specific
numbers for the four values x, y, z and w, is called a STATE
VECTOR and completely specifies the state of the system. The set
of all possible state vectors (in this case a 4-dimensional space)
is called the STATE SPACE of the system. As the system transitions
from state to state within the state space, it traces a path called
a TRAJECTORY of the system.
Using matrix algebra, it becomes possible to find solutions to
many such linear systems of ODEs.
USING SPREADSHEETS TO IMPROVE YOUR INTUITION ABOUT MATRIX ALGEBRA
Dan Bricklin has spoken of watching his university
professor create a table of calculation results on a
blackboard. When the professor found an error, he had
to tediously erase and rewrite a number of sequential
entries in the table, triggering Bricklin to think that
he could replicate the process on a computer, using the
blackboard as the model to view results of underlying
formulas. His idea became VisiCalc, the first application
that turned the personal computer from a hobby for computer
enthusiasts into a business tool.
VisiCalc went on to become the first "killer app", an
application that was so compelling, people would buy a
particular computer just to own it. In this case the
computer was the Apple II, and VisiCalc was no small
part in that machine's success.
-- Wikipedia entry for "spreadsheet"
( http://en.wikipedia.org/wiki/Spreadsheet )
When I first tried to learn matrix algebra I found it baffling.
Everything seemed trivially simple right up to the point where
it became completely opaque. Again this is probably because
of the way it is taught. Imagine if you signed up for a wood-
working class and spent months making a chisel, awl, plane, drill,
adz, and file without ever using any of them, or even being
told what they are for. But you sure knew how to sharpen
them! Matrix algebra comes with a similar set of sharp tools,
and most classes I've taken and texts I've read that cover the
material concentrate on tool-sharpening at the expense of
building birdhouses (to stretch the metaphor).
One of the first tools you learn about is the inscrutable
DETERMINANT. It begins with a simple definition for the 2D
case. As the Wikipedia entry for determinant
( http://en.wikipedia.org/wiki/Determinant )
so succinctly says:
The 2 x 2 matrix A =
| a b |
| c d |
has determinant det(A) = a*d - b*c
The the determinant of a 3x3 matrix B
| a b c |
| d e f |
| g h i |
is defined as (take a breath) for ANY ROW the sum of each element
times the determinant of the sub-matrix formed by taking all the
elements which it doesn't share a row or column with.
In other words:
a*det(P) + b*det(Q) + c*det(R)
where P =
| e f |
| h i |
Q =
| d f |
| g i |
and R =
| d e |
| g h |
And it gets gnarlier from there. Note that I used the top row
but you can use ANY ROW. This is part of the magic of linearity.
Also note that you can take any linear combination of rows and
replace any or all rows with them, and the determinant of the
resulting matrix will be the same.
For example, take A =
| 3 2 |
| 6 4 |
the determinant is 3*4 - 2*6 = 0.
If we replace the top row with 5 times itself, we get A* =
| 15 10 |
| 6 4 |
so det(A*) = 15*4 - 6*10 = 0. Bingo! Linear combinations of
linear things are generally equivalent.
In attempting to "get my head around" matrix algebra during that
oft-mentioned UCLA course, I found it useful to implement some
matrix operations using a spreadsheet program, like Microsoft Excel.
I've replicated a few of these and posted them for you to play with.
The first computes determinants of both a 2x2 and 3x3 matrix.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/det2D3D.xls )
The next performs a rotation about the Z axis on a 3D vector.
The 3x1 position vector is multiplied by a 3x3 rotation matrix
to produce a 1x3 position vector. (In this one I churned out
a few positions for a rotated point and plotted them as well.)
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/rot.xls )
And lastly I compute a general multiplication of a 3x1 and 3x3
matrix to produce a 1x3 vector.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/vA.xls )
If you punch different numbers into the inputs and watch how the
outputs change, it will help develop your intuition for matrix
operations; even more so if you create your own spreadsheets.
EIGENVECTORS
In mathematics, an EIGENVECTOR of a transformation is a vector
whose direction is unchanged by that transformation. The factor
by which the magnitude is scaled is called the EIGENVALUE of
that vector. Often, a transformation is completely described
by its eigenvalues and eigenvectors. The EIGENSPACE for a
factor is the set of eigenvectors with that factor as eigenvalue.
In the specific case of linear algebra, the EIGENVALUE PROBLEM
is: given an nxn matrix A, do there exist nonzero vectors x in
R^n such that Ax is a scalar multiple of x? If so: the scalar
is denoted by the Greek letter lambda and is called an EIGENVALUE
of the matrix A, while x is called an EIGENVECTOR of A
corresponding to lambda and the following has one or more
solutions: Ax = lambda x.
These concepts play a major role in several branches of both
pure and applied mathematics -- appearing prominently in linear
algebra, functional analysis, and to a lesser extent in
nonlinear situations.
-- Wikipedia entry for "Eigenvalue, eigenvector and eigenspace"
( http://en.wikipedia.org/wiki/Eigenvector )
So, did that quote from Wikipedia make sense? We are starting with
a NxN matrix, which -- in its simplest interpretation -- is a set
of linear rules for transforming an N-vector into another N-vector.
What we want to know is: does an input vector exist which is ONLY
SCALED when it is transformed by the matrix?
There are techniques for determining if the eigenvectors exist,
and if so finding them. These techniques use determinants from
matrix theory and algebraic tools like the quadratic equation.
( http://tutorial.math.lamar.edu/AllBrowsers/3401/LA_Eigen.asp )
But what does it MEAN? So what if there are vectors that are
only scaled by a matrix? These are the places in the state space
that have their ANGLE UNCHANGED by the matrix. The Wikipedia
entry for "Eigenvalue, eigenvector and eigenspace"
( http://en.wikipedia.org/wiki/Eigenvector )
has an illustration that is useful: the Mona Lisa under a "shear
transform" (sometimes called "scissoring" or "racking") showing
two eigenvectors which maintain their angles.
I thought about the conformal maps I'd visualized, and which
ones could be done with a 2x2 real matrix operating on a 2-vector,
instead of algebra on a complex value s.
(To review 2D matrix algebra: if input is |x, y| and matrix is
| a b |
| c d |
then |a*x + c*y, b*x + d*y| is the output.)
IDENTITY of course,
g(s) = s
can be done with the matrix:
| 1 0 |
| 0 1 |
In this matrix every vector is an eigenvector with eigenvalue = 1.
And the SCALE function,
g(s) = s*r
where r is a real number constant, is performed by the matrix:
| r 0 |
| 0 r |
Also in this matrix every vector is an eigenvector with
eigenvalue = r.
I puzzled a while over ROTATE:
g(s) = s*j
(where j is the imaginary quantity). Its matrix is:
| 0 -1 |
| 1 0 |
Solving for the eigenvalue I found the quadratic equation
had a negative number under the radical (square root sign),
so there were no real roots.
That makes sense, because only the point at {0 0} is
untransformed by the matrix, being the center of rotation,
and that eigenvector is specifically disallowed by the warranty.
The others, TRANSLATE:
g(s) = s + c
SQUARE:
g(s) = s^2
and INVERSE:
g(s) = 1/s
are all non-linear and cannot be expressed with a 2x2 matrix.
As I played with this concept, I asked myself what its deeper
meanings were. What did the name mean? German for "self-vector."
Talking to friends about it, I realized perhaps a better
translation would be "ego vector." Sort of a "Triumph of the
Will" concept. Under the slings and arrows of the transformation
matrix only the powerful ego vectors maintain their direction!
Also, I should note that if you type "eigenvector" into Google Images
( http://images.google.com )
you get some very interesting pictures.
STABILITY
An example of an unstable control system is the automatic
temperature control of an electric blanket with dual controls,
where the husband and wife each has the wrong temperature
control. As the wife selects a higher desired temperature,
the extra heat is applied to the husband, who reduces the
temperature setting on his controller. However, this action
lowers the temperature of the portion of the blanket on his
wife, who in turn selects an even higher temperature on her
controller. This process continues until the wife's side of
the blanket is completely off and the husband's side of the
blanket is at maximum temperature. In this example the
controller quantity, temperature, moved in opposite direction
to the desired value, and thus it represents an unstable system.
-- Roberto Saucedo and Earl E. Schiring, 1968
"Introduction to Continuous and Digital Control Systems"
( http://www.amazon.com/exec/obidos/ASIN/B000H4H4WG/hip-20 )
One of the most common things an engineer needs to determine of
a mathematical model of a system is its STABILITY.
If you are using the old classical approach of transfer functions,
you find a RATIONAL TRANSFER FUNCTION of the form:
X(z) = P(z)/Q(z)
and then determine when P(z) = 0 (the ZEROS of the system)
and Q(z) = 0 (the POLES of the system) and plot them in the
complex plane as a POLE-ZERO PLOT.
( http://en.wikipedia.org/wiki/Pole-zero_plot )
As Wikipedia explains:
The region of convergence for a given transfer function
is a disk, punctured disk, or annulus which contains
no poles.
If the disc includes the unit circle, then the system
is BIBO stable.
The referenced article on BIBO stability
( http://en.wikipedia.org/wiki/BIBO_stability )
defines:
BIBO Stability is a form of stability for signals and
systems. BIBO stands for Bounded Input/Bounded Output.
If a system is BIBO stable then the output will be bounded
for every input to the system that is bounded. (A signal
is bounded if there is a finite bound B > 0 such that the
signal magnitude never exceeds B.)
This is a system that is guaranteed not to blow up if you don't
feed it an already-blown-up signal.
To tell you the truth this classical stuff seems like so much
voodoo to me, operating in a FREQUENCY DOMAIN that I have never
understood very well. But luckily, here in the digital age,
we don't need to do that very much any more.
Instead, using the state space approach, we express our systems
as ODEs, then express the ODEs in matrix form, and then -- guess
what? -- find the EIGENVECTORS and EIGENVALUES of the system!
I got the following from "Nonlinear Ordinary Differential Equations"
(book, 1977) by D. W. Jordan and P. Smith.
( http://www.amazon.com/exec/obidos/ASIN/0199208247/hip-20 )
(Why am I providing a linear solution from a book on non-linear
equations? Well, it was near the front of the book, in a section
on linearly approximating nonlinear systems.)
[Given:]
x' = a*x + b*y, y' = c*x + d*y
It is known that there are nontrivial solutions of [these
equations] of the form:
x = r*e^(lamda*t), x = s*e^(lamda*t)
where r and s are related constants.
Lambda, of course, is an eigenvalue of the system (usually
there are two). It is found as roots of the CHARACTERISTIC EQUATION:
lambda^2 - (a + d)*lambda + (a*d - b*c) = 0
Now, the tricky thing is that sometimes these roots are complex.
What? Eigenvalues can be complex?
Why, yes, and that's when it gets interesting.
( http://www.sosmath.com/matrix/eigen3/eigen3.html )
In the last section I puzzled over a rotation matrix:
| 0 -1 |
| 1 0 |
explaining that:
Solving for the eigenvalue I found the quadratic equation
had a negative number under the radical (square root sign),
so there were no real roots.
If I had continued anyway, I would've found two complex roots:
lambda = j
and:
lambda = -j
which makes possible a CIRCULATING ATTRACTOR, or an OSCILLATOR.
It also enables a SPIRAL ATTRACTOR as explained in an on-line
essay on "Systems With Complex Eigenvalues."
( http://ltcconline.net/greenl/courses/204/Systems/complexEigenvalues.htm )
(Scroll down to phase portrait for a picture of the attractor.)
As explained in the on-line essay "The PhasePlane for a Linear system"
( http://www.math.pitt.edu/~bard/classes/xppfast/lin2d.html )
the locations of eigenvalues characterize the behavior of the system:
[Looking] at the phaseplane for the two-dimensional linear system
of differential equations:
x' = ax + b y
y'= cx + dy
We know that the behavior of this system is completely determined
by the eigenvalues of the matrix A whose entries are a,b,c,d.
These are the normal possibilities:
* Saddle point -- one positive and one negative
eigenvalue
* Unstable node -- two positive real eigenvalues
* Stable node -- two negative real eigenvalues
* Unstable vortex or spiral -- complex eigenvalues
with positive real parts
* Stable vortex or spiral -- complex eigenvalues
with negative real parts
In addition there are a number of degenerate cases:
* Center -- a pair of pure imaginary eigenvalues
* Degenerate node -- two identical eigenvalues
* Line field -- a zero eigenvalue
Once you allow this enhancement, you get the new-school definition
of stability: A LINEAR SYSTEM IS STABLE IF ITS EIGENVALUES HAVE
NEGATIVE REAL PARTS.
( http://en.wikibooks.org/wiki/Control_Systems/State-Space_Stability )
Since solutions are of the form k*e^(lambda*t), and recalling
that e to a complex power sigma + omega*j is defined:
e^(sigma + omega*j) = e^sigma*(cos(omega) + j*sin(omega))
Since we know a cosine is going to stay safely between -1 and 1,
clearly sigma must be negative to keep the result from expanding
without limit exponentially over time.
In the on-line course notes for "MAS375: MODELLING AND SIMULATION"
by Dr. Mark Lukas at Murdoch University, Perth, Australia,
( http://www.maths.murdoch.edu.au/units/mas375/unitnotes/unitnotes.pdf )
chapter 3, starting on page 23, has some interesting illustrations
of software solutions to system equations.
These days most on-line information about using eigenvalues to
determine stability are part of course notes -- and especially lab
notes -- involving using modelling software, such as Matlab or
Mathematica to study systems theory. The on-line essay "Linear
System of Differential Equations" from the University of Massachusetts
Dartmouth shows solutions using the program TEMATH.
( http://www2.umassd.edu/temath/TEMATH2/Examples/LinearSystemsOfDiffEqs.html )
This is a great way to learn this stuff, and I recommend it.
CONTROL
It's control. All these things arise from one difficulty:
control. For the first time it was inside, do you see.
The control is put inside. No more need to suffer passively
under 'outside forces' -- to veer into any wind.
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel)
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
So far, since the section entitled "THE LINEAR CASE," we have been
looking at the ANALYSIS side of control theory. But eventually an
engineer must eventually move to the DESIGN side. This means taking
a system that isn't behaving as desired and ADDING COMPONENTS (usually
involving feedback) to create a NEW SYSTEM with the desired behavior
in its stable modes. For example, adding a governor to a steam engine.
The on-line publication "Design of Simple Digital Controllers"
(August 1996) by Ming T. Tham of the Department of Chemical and Process
Engineering at the University of Newcastle upon Tyne, UK,
( http://lorien.ncl.ac.uk/ming/digicont/control/digital1.htm )
summarizes the process.
One of the limiting trade-offs that comes up in control system
design is accuracy versus stability. As Saucedo and Schiring
( http://www.amazon.com/exec/obidos/ASIN/B000H4H4WG/hip-20 )
explain:
Accuracy
... In the open-loop system the output ... depends on [the
input] completely. Any imperfections or changes in [the input]
from the nominal condition result directly in output voltage
inaccuracy. However, for the feedback configuration, a change
in the output from a desired value is reflected in an error
signal ... The error signal causes a change in the output
in a direction to bring the output toward the desired voltage.
Thus changes in the output are detected and corrected by the
feedback has greater accuracy. ...
Stability
Most physical systems are inherently stable open-loop, but
the addition of feedback can cause the closed-loop system to
be unstable. In fact, the greater the final accuracy desired,
the less stable the system becomes. For high accuracy, the
gain associated with [the input] should be high, so that even
the smallest detectable error signal may be amplified and
produce a correction to the output. However, for high
values of forward path gain, the corrective action produced
at the output can be too large, with resulting overshoot or
undershoot of the output from its original offset condition.
The error then reverses sign, and the corrective action also
reverses. If the gain is too large, the output can start
oscillating with either sustained or increasing amplitude,
obviously an unstable situation. Thus the features of
accuracy and stability are opposites in the sense that
increasing accuracy decreases stability, and vice versa.
In other words, the goals of EXACT REPRODUCTION and NOISE REJECTION
are in basic conflict.
Saucedo and Schiring also talked about "deadbeat response systems."
These are systems in which the specified controller response is at
discrete time steps; the system requirements specify values only
at time "ticks" in a sample-data stream.
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/ICDCS0002.jpg )
It turns out these systems have an odd failure mode, as in an example
of a time-sampled controller for a missile launcher:
...the output sequence agrees with the actual launcher movement
(at the sampling instants) and behaves as predicted. In between
samples, however, the launcher oscillates severely. It does
little good to point the launcher quickly and accurately at a
hostile target and settle to zero steady-state error at the
sampling instants and yet the remainder of the time, which is
in the majority, to wave it haphazardly at the heavens. These
oscillations, which exist in the real world and are not
predictable by use of z transforms (and indeed are not seen
by the digital computer), are called HIDDEN OSCILLATIONS. They
hide behind the multitude of [terms].
...
This is characteristic of deadbeat response systems: they are
highly tuned to a specific input function.
...
Moreover, these high-frequency ripples, or hidden oscillations,
would surely excite the mechanical resonances stated previously.
The deadbeat idea, however, is intriguing and warrants further
investigation.
But it occurs to me that in the digital age, it might be possible
to simulate such a super-sensitive controller while monitoring for
instability, and switching to another, more stable technique (probably
involving more time samples) when necessary.
Indeed, a Google search turned up a commercial offering that
uses deadbeat techniques in chemical engineering and process control.
( http://www.gossenmetrawatt.com/english/seiten/licenceforuseofdead-beatpdpicontrolalgor.htm )
SHAKEN NOT STIRRED
Finally, there is a physical problem that is common to
many fields, that is very old, and that has not been
solved. It is the analysis of circulating or turbulent
fluids. The simplest form of the problem is to take a
pipe that is very long and push water through it at high
speed. We ask: to push a given amount of water through that
pipe, how much pressure is needed? No one can analyze it
from first principles and the properties of water. If the
water flows very slowly, or if we use a thick goo like honey,
then we can do it nicely. You will find that in your textbook.
What we really cannot do is deal with actual, wet water
running through a pipe. That is the central problem which
we ought to solve some day, and we have not.
-- Richard Feynman, 1963
"The Feynman Lectures On Physics"
( http://www.amazon.com/exec/obidos/ASIN/0201021153/hip-20 )
quoted in "Turbulence in Nature and in the Laboratory"
by Z. Warhaft
( http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=128565 )
Okay, deep breath. MOST OF THE ABOVE INFORMATION IS, IN THE
GENERAL CASE, WRONG. It applies only to the linear cases. Many
control engineers have spent careers modelling, analyzing and
controller linear systems, only to have intuitive problems with
the nonlinear ones.
So how do you attack the nonlinear systems? Well, as it turns out,
one fruitful approach has been behind the door marked "unexplained
phenomena."
In the early eighties, following a review in the CoEvolution
Quarterly,
( http://en.wikipedia.org/wiki/CoEvolution_Quarterly )
I acquired a copy of "Sensitive Chaos: The Creation of Flowing
Forms in Water and Air" (book, 1965) by Theodor Schwenk.
( http://www.amazon.com/exec/obidos/ASIN/1855840553/hip-20 )
This argument for a living universe from a Theosophy point of view
relies mostly on photographs of living creatures compared with
swirling water to argue that ALL WATER HAS CONSCIOUSNESS. Well,
I don't know about that, but I enjoyed the pictures, and I began
thinking of the book as a gallery of unsolved problems in fluid
dynamics.
( http://jonathanmackenzie.net/aeoc/schwenk.htm )
Almost a decade later when I worked for Stellar helping to sell
graphics supercomputers to scientists, I accepted every invitation
to speak I received, and one of them was from USC's honors engineering
students. I prepared a set of overheads that began with a slide
from Schwenk's flowing, turbulent water pictures. "Does anyone
know what this is?" I asked. Someone guessed, "Turbulence."
"That's right," I said. The I showed a slide of the Cantor Set.
( http://en.wikipedia.org/wiki/Cantor_set )
"Does anyone know what this is?" I again asked. "Cantor dust,"
another student said.
"Very good."
As Wikipedia explains:
The Cantor set is created by repeatedly deleting the open middle
thirds of a set of line segments. One starts by deleting the
open middle third from the interval [0, 1], leaving two line
segments: [0, 1/3] and [2/3, 1]. Next, the open middle third of
each of these remaining segments is deleted. This process is
continued ad infinitum. The Cantor set contains all points in
the interval [0, 1] that are not deleted at any step in this
infinite process.
I then suggested that what these two had in common, the observed
turbulent flow and the theoretical "fractal" from 1884, was non-
linearity.
At the time, by hanging out with scientists doing Computational
Fluid Dynamics (CFD) with supercomputers, I was learning to
think a new way about cream poured into a clear glass of iced coffee,
or of cigarette smoke rising under a lamp's light in a black and
white movie, or of meandering rivers depositing silt at the
meanders as they slow to make the turn, and then extending the
meander farther out to get around the silt, deepening the channel
on the outside bank, or even cuckoo clocks hung on the same wall
synchronizing their pendulums.
THE NON-LINEAR CASE
But it was a neat theory, and he was in love with it. The only
consolation he drew from the present chaos was that his theory
managed to explain it.
-- Thomas Pynchon, 1963
"V." (novel)
( http://www.amazon.com/exec/obidos/ASIN/0060930217/hip-20 )
I had a professor once who said "everything is linear if you break
it down." I said, "Yeah, and everything's dead if you kill it."
One of the most common methods used to study nonlinear systems is to
try and "linearize" them somehow. You do this when you have to, I
suppose, but it seems to me you will miss all the non-linear weirdness
this way. It's like if you only studied lungfish when they were on
land, you would conclude they acted a lot like land animals. You
would report that they never swim or breathe water.
The reason everybody wants to do this, of course, is because the
nonlinear equations are UNSOLVABLE. But they cannot be denied.
By hook or by crook we have to do something with them.
From the 1960s to the 1980s there was a paradigm shift in mathematical
physics that resulted in the emergence of CHAOS THEORY, when researchers
finally began to notice and document all the bizarre behaviors that
nonlinear systems could exhibit: STRANGE ATTRACTORS such as the
Lorenz Butterfly,
( http://en.wikipedia.org/wiki/Lorenz_attractor )
Rossler Bands,
( http://hypertextbook.com/chaos/eyecandy/strange-rossler.html )
and others:
( http://sprott.physics.wisc.edu/fractals/cat00002.gif )
( http://sprott.physics.wisc.edu/fractals.htm )
Of course later it was figured out that some of the best tools
for dealing with these beasts were developed by Russians in the
1930s, such as the Lyapunov Exponent.
( http://en.wikipedia.org/wiki/Lyapunov_exponent )
Wikipedia also has this connection with eigenstuff:
Whereas the (global) Lyapunov exponent gives a measure for
the total predictability of a system, it is sometimes
interesting to estimate the local predictability around a
point x0 in phase space. This may be done through the
eigenvalues of the Jacobian matrix J^0(x0). These eigenvalues
are also called local Lyapunov exponents.
These new (and newly-rediscovered) models began to link up with
some of the old unexplained phenomena, such as John Scott Russell's
1834 discovery of the WAVE OF TRANSLATION (later named the SOLITON),
first seen as a long-distance traveling wave on a canal, or rogue
waves on the oceans long reported by sailors and pooh-poohed by
scientists.
I have previously written of the numerical experiments by Fermi,
Pasta and Ulam in the 1950s
( http://www.osti.gov/accomplishments/pdf/A80037041/A80037041.pdf )
and Wolfram in the 1990s
( http://www.wolframscience.com )
investigating nonlinear systems through simulation.
One driving force for the new science of COMPLEXITY being proclaimed
has been a desperate need for new models in economics. John Reed was
CEO of Citibank with about $300 billion in debt owed by Third World nations
on his books that was looking pretty shaky. But his economists, using
linear "equilibrium" based computer models, told him that default was
impossible. You know what happened. In the aftermath Reed helped fund
and found the Santa Fe Institute to research nonlinear economic models
(among many other things).
This is described in "Complexity: The Emerging Science at the Edge of
Order and Chaos" (book, 1992) by Mitchell M. Waldrop.
( http://www.amazon.com/exec/obidos/ASIN/0671872346/hip-20 )
Any early approach to controlling chaotic systems was to drive
them out of the chaotic region, into a near-linear mode, and
then use linear techniques. This didn't always work so well.
In his landmark paper "Controlling Cardiac Chaos" (1992)
(Science, Vol 257, Issue 5074, 1230-1235),
( http://www.sciencemag.org/cgi/content/abstract/257/5074/1230 )
Dr. Alan Garfinkel and his colleagues showed how to control a
chaotic heartbeat without leaving the chaotic region, merely by
understanding it (and doing real-time digital computations to
know how to regulate it).
Recent work in the U.S. and Japan on non-linear N-body celestial
mechanics has found energy-saving chaotic trajectories from the
earth to the moon, for example.
EVERYTHING HAS TO GO SOMEWHERE
1) Nature is more complex than we know, and probably more complex
than we can know. 2) Everything has to go somewhere. 3) There is
no such thing as a free lunch. 4) Nature knows best.
-- Barry Commoner, 1971
"The Closing Circle"
( http://www.amazon.com/exec/obidos/ASIN/0553202464/hip-20 )
I suspect some of you may have voted for this topic because you thought
it would be about ecology a la Barry Commoner's "Everything has to go
somewhere." No such luck. Sorry I didn't mention it sooner.
For me the phrase is a sort of shorthand for some lesson's I've
learned about the geometry of systems theory. (After all,
topology was invented to analyze nonlinear systems.)
If you have a blob of something and you want to minimize its surface
area, what shape must it form? A sphere of course. But what if you
want to MAXIMIZE its surface area? There's really no right answer,
without additional constraints. Some sort of fractal is required,
certainly. The air cooling fins on a motorcycle engine are a step
in the right direction. A tree filled with leaves seems to be
solving a similar problem, or maximizing sun-catching surface for
unit of resources.
I think of this when I look at the cigarette smoke under the 1940s
lampshade in a noir genre movie. If the smoke column were attempting
to MINIMIZE its length it would be a straight line. But the smoke
is very hot, and so is expanding relative to the cooler air around it.
It needs to stretch, to "go somewhere," and so it forms the twists
and gnarls that cinematographers love so much.
There is one fact which seems to have an overriding influence on
system trajectories, and that is that, BY DEFINITION, the STATE
of the system (its current state vector in the state space) is ALL
YOU NEED TO KNOW in order to predict its future behavior. If not,
there is a HIDDEN VARIABLE and you don't have the complete state.
But if you DO have the complete state specified, then if the
system ever returns to a state it has been in before, then it is
now in a so-called INFINITE LOOP and will surely LOOP FOREVER.
So, think about this. A system at any instant will transition
either to:
a state it has never been in before, or
a state it HAS been in before, and now it's in an infinite loop.
So suppose you have your trusty lab computer observe some nonlinear
systems for a while, and throw away any observations of loops --
oscillations and repetitions. Eventually you look at the data to find
a whole zoo of ways to "not loop." The system keeps having to go
someplace "new" in order to not loop. In a Darwinian fashion,
chaos has been selected for.
We have found that in two dimensions chaos is impossible; attractors
can only be topological distortions of points, lines fleeing to infinity,
circles, and spirals (in and out). But in three dimensions and higher
the "strange" attractors keep finding "new places to go."
I keep thinking their is further weirdness waiting for us in the
higher dimensions.
IMPROVING THE QUALITY OF COMMON SENSE
Ask the Germans especially. Oh, it is a real sad story, how
shoddily their Schwarmerei for Control was used by the folks
in power ... "Paranoid Systems of History" ... has even
suggested ... that the whole German Inflation was created
deliberately, simply to drive young enthusiasts of the
Cybernetic Tradition into Control work ... If any of the young
engineers saw correspondence between the deep conservatism of
Feedback and the kinds of lives they were coming to lead in the
very process of embracing it, it got lost, or disguised --
none of them made the connection...
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel)
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
An enterprising turkey gathered the flock together and,
following instructions and demonstrations, taught them
how to fly. All afternoon they enjoyed soaring and flying
and the thrill of seeing new vistas. After the meeting,
all of the turkeys walked home.
-- Merlin R. Lybbert
"Ensign" May 1990, p. 82
So, what's the point of these mental exercises? I would
argue that it is to improve your intuition, gentle readers.
I don't suppose that very many of you are going to become
control engineers and try to figure out the harmonic modes
of analog electrical circuits, or design mechanical
oscillators using spring, dashpots and motors, so the
specific knowledge of zero-pole plots and eigenvectors
won't be of much to use to you. But all of us deal with
complex systems, and a better "feel" for how these systems
CAN behave would do us all some good.
I remember in his book "Cybernetics" (1948)
( http://www.amazon.com/exec/obidos/ASIN/026273009X/hip-20 )
Norbert Wiener describes how a fire-control system which he designed
during WWII keeps an anti-aircraft gun aimed at the point in the sky
where an airplane is EXPECTED to be by the time the bullets reach it;
all the operator has to do is keep the gun sight aimed at the aircraft's
CURRENT position. If the operator doesn't move the sight for a while
-- or wanders off -- the gun stays pointed at a fixed point. The fire
control system "thinks" the plane has no relative sideways motion
-- i.e. is approaching or receding -- and so aims straight at it.
Wiener points out that if you slap your hand against the gun, it feels
solid, like it is mounted in concrete. But that's just the fire
control system responding to feedback quickly with an opposite force.
If you switch off the power, the gun falls passively to the deck.
From this I learned that whatever feels solid in fact has a
feedback system behind it. Even if a pole IS mounted in concrete
there is a feedback system behind it; a solid object anchored to
the Earth has the whole planet to help it push back when pushed.
Another thing I've learned from systems theory is the importance
of understanding time-lag effects. Attempts to control systems
can paradoxically drive them into instability. Recall trying
to adjust the temperature of a shower with a long lag time between
turning the faucet and feeling the temperature change. I remember
when I was a freshman in college I wanted to be a high school
math teacher, but at a meeting for education majors I was warned
of an impending teacher "glut," and changed my plans. By the
time I was senior there was a huge teacher shortage. The education
department didn't understand time lag effects.
Still another thing I've learned from systems theory is that what
seems paradoxical when analyzed from a physics perspective can
make sense when explained with systems theory.
For example, various collections of "Murphy's Laws" often include
the paradoxical "law" that:
Any stone in your boot always migrates against the pressure
gradient to exactly the point of most pressure.
A systems theory explanation for this effect is that the point
of highest pressure gradient pinches the tightest. During hiking,
footsteps create an oscillating pressure field, in which stones
can rattle around. This they do until they are pinched so tight
that they can't move. The "point of most pressure" is therefore
an ATTRACTOR.
In his essay "Cybernetic Explanation" -- reprinted in "Steps To An
Ecology of Mind" (book, 1972) --
( http://www.amazon.com/exec/obidos/ASIN/0226039056/hip-20 )
Gregory Bateson says:
Causal explanation is usually positive. We say that billiard
ball B moved in such and such a direction because billiard ball
A hit it at such and such an angle. In contrast to this,
cybernetic explanation is always negative. We consider what
alternate possibilities could conceivably have occurred and
then ask why many of the alternatives were not followed, so
that the particular event was one of the few which could, in
fact occur...
In cybernetic language, the course of events is said to be
subject to restraints, and it is assumed that, apart from such
restraints, the pathways of change would be governed only by
equal probability.
These are lessons that control engineers (sometimes) learn, and
I would like more people to have available, especially those
who make a large difference.
We need more systems theory understanding among leaders in
government (and their electorates), not just among the
engineers who design the guidance systems for the weapons
they command.
BUZZ BOMBS
Remittance men from all over the world will come to
Heidelberg before long, to major in guilt. ... Sorry
-- not for Achtfaden here, shrugging ... -- he only worked
with [the V2 rocket] up to the point where the air was
too thin to make a difference. What it did after that
was none of his responsibility. Ask ... the re-entry
people. Ask the guidance section, they pointed it
where it was going. . . .
-- Thomas Pynchon, 1973
"Gravity's Rainbow" (novel)
( http://www.amazon.com/exec/obidos/ASIN/0140188592/hip-20 )
As we saw above, it seems creepily easy to open a control theory
text, like Saucedo and Schiring, and encounter -- as if it were
nothing special -- a sentence such as:
It is desired to design a missile launcher control system
to seek and follow a hostile target in a prescribed manner.
Often there is an accompanying diagram of the missile launcher
blowing stuff up "in a prescribed manner."
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/ICDCS0001.jpg )
So, you want me to help improve the efficiency of a machine for
bringing agonizing death to humans? Without a thought as to the
moral implications? Oh, we're just going to use it as an example
just to learn something about Truxal's Method (or whatever), so
LATER we can improve actual missile launchers?
Sooner or later every cyberneticist runs into the war work issue.
My story is far from unique.
I remember in the late 1970s the two scariest new phrases I learned
were: "acquired immune deficiency syndrome," and "cruise missile."
I think the scariness in the concept of AIDS is pretty obvious.
Almost as obvious is the danger of an undetectable ("stealth")
delivery system for nuclear weapons. (ICBMs, in contrast, are
detectable but mostly unstoppable.) There is the danger of the
technology "falling into the wrong hands." There is also the danger
of power corrupting -- politicians tend to use the weapons they are
given.
At one point in my career I attempted to refuse to work on weapons
projects. I took a job in 1979 reluctantly with a "computer output
to microfilm " (COM) company called Datagrafix, knowing the parent
company, General Dynamics (isn't that a great name?) made Tomahawk
Cruise Missiles.
( http://en.wikipedia.org/wiki/BGM-109_Tomahawk )
I thought I'd be insulated from that side of the business. But
during my first week on the job, I was sent to the General Dynamics
facility in the Kearny Mesa neighborhood of San Diego for a training
class, the same facility where the Atlas ballistic missiles were
built in the 1950s and '60s,
( http://www.globalsecurity.org/space/facility/atlas_f.htm)
and there on the wall of the classroom was a diagram of the
Tomahawk.
( http://www.ausairpower.net/Tomahawk-Subtypes.html )
Arg! I stayed for six months before fleeing. (The company
had other dysfunctions besides making robot kamikazes.)
A few years later I found myself at a company making a pioneer
3D graphics system, the Poly 2000 by GTI Corp. "The applications
were endless" we heard during the R&D phase, but when we had a
product ready to sell we found out our only viable market was
military simulators.
Arg! I ended up fleeing that company as well, for several reasons,
and I ended up getting job offers from two customers who'd bought
the 3D graphics systems: Hughes Aircraft Company in El Segundo,
researching combat helicopter design, and Rockwell International in
Downey, chasing the primary NASA contract for building the space
station Reagan had announced in 1983. I chose the civilian space
project over the weapons project.
But then at Rockwell the civilian business got slow and I was
assigned to do 3D graphics for Reagan's "Star Wars" program, the
Strategic Defense Initiative (SDI), which he also also announced
in 1983. This crash program was recommended by good old Dr. Edward
Teller, "Father of the H-Bomb." (George Gamov teased him with a
thought experiment in which he shook hands with "Dr. Anti-Teller"
and was annihilated in a burst of gamma rays.)
( http://en.wikipedia.org/wiki/Edward_Teller )
The goal of SDI was to stop those Soviet ICBMs -- moving at
supersonic speeds and armed with Multiple Independently-targetable
Re-entry Vehicles (MIRVs) with H-bomb warheads -- in mid-flight.
( http://en.wikipedia.org/wiki/MIRV )
My conscience was troubled. I kept working, but was plagued by a
series of bicycle accidents culminating in the one that totaled my
bike and put me in the hospital. My unconscious mind was trying
to tell me something. My life was incongruent. My values didn't
match my actions. I did some soul searching, and I realized that
if I thought weapons work was wrong, refusing to participate wasn't
good enough. They were taking money out of my paycheck every month
to build this stuff no matter where I worked. No personal boycott
was going to make a whit of difference.
I made some decisions:
1) I was being too hard on myself. I have a right to
support my family. I needed to forgive myself for
war work.
2) If I wanted to make a difference, I needed to WORK
WITHIN THE SYSTEM to make a difference.
3) I still didn't LIKE to do war work, and would avoid
it when I could.
My next career move was to a sequence of companies that sold
tools for supercomputing and scientific visualization. They
seemed like generic scientific tools, but somehow our best customers
were designing stealth aircraft or H-bomb detonators at places
like the Lockheed Skunkworks
( http://en.wikipedia.org/wiki/Lockheed_Skunkworks )
or Sandia National Labs.
( http://en.wikipedia.org/wiki/Sandia_National_Laboratories )
But I was patient and my opportunity to make a difference came.
I helped the company land some large commercial accounts in
medical imaging and integrated circuit design, and prepared
for the day in 1989 when the Berlin Wall fell and all the
aerospace engineers said "Oh s###!, now what do we do?"
It was one of the time in my life I felt the centrifugal
force from being close the the "hinge of history."
My company was able to survive in the post-cold war and
post-Soviet Union world, partly due to my efforts,
and all over America there were similar "sword into
plowshare" moves that I believe ending up giving our
economy an uplift.
But before that I took my wife on vacation in New Mexico, and
she asked to visit the National Atomic Museum at Sandia Labs.
( http://www.atomicmuseum.com )
and we saw a craft that was labeled "America's first cruise
missile." I'm pretty sure it was a North American X-10.
( http://www.fas.org/nuke/guide/usa/icbm/n19980710_981014.html )
As the Boeing web site
( http://www.boeing.com/phantom/xplanesdt.html )
describes it:
As part of the Navaho missile program, the X-10 test drone
was essentially the first cruise missile. The excessive
weight of nuclear warheads at the time initiated the
investigation of this unusual delivery system. From its
first flight on October 14, 1953, the X-10 was controlled
by either a pilot on the ground or from the backseat of an
ET-33 chase plane. The X-10 program fostered a multitude of
contributions into follow-on systems.
Staring it at it, I said, "Of course, the real 'first cruise
missile' was the buzz bomb -- the V-1."
( http://en.wikipedia.org/wiki/V-1_flying_bomb )
Wikipedia summarizes:
The Vergeltungswaffe-1, V-1, ... known colloquially in
English as the Flying bomb, Buzz bomb or Doodlebug, was
the first guided missile used in war and the forerunner
of today's cruise missile.
Of course it was launched by the Germans against the English.
It was really just a little robot airplane with a bomb aboard.
I heard that they'd buzz as the flew, and when they ran of fuel
they'd STOP buzzing, and that's when you'd dive for cover,
because they'd fall to the ground and explode.
I saw that the development of guided weapons was gradual and
probably unstoppable. "Cruise Missile" was a new marketing term
for something in the works since archery. I realized I'd made
my peace with these war machines. When the Tomahawk was used
in Gulf Wars I, II and III to save American lives, but only
strategic and sparingly because they are REALLY EXPENSIVE
(after the radars are knocked out you send in planes) I
actually bought a Tomahawk T-shirt at a local air show.
I still hate war, but I don't oppose the soldiers or their
gadgets, just the political use of war as a proactive tool.
I apply my pressure directly through my vote and political
contributions and actives, mostly involving the Libertarian
Party.
( http://en.wikipedia.org/wiki/Libertarian_Party_(United_States) )
But others have not taken this trajectory. The founder and coiner
of cybernetics, Norbert Wiener, after doing war work for
World War II, wrestled with his conscience in the 1950s at MIT
and ended up rejecting all government funding, in contrast to
most of his colleagues, who were cooking all sorts of ways to
get Army funding. The Lincoln Lab was born in this climate.
This story is told in the recent biography "Dark Hero Of The
Information Age: In Search of Norbert Wiener The Father of
Cybernetics" (2004) by Flo Conway and Jim Siegelman.
( http://www.amazon.com/exec/obidos/ASIN/0738203688/hip-20 )
Bucky Fuller loved the Navy and the education he got there,
but he hated war. He wrote against building weaponry with
technology, encouraging "livingry" instead, such as cheap
mass-produced shelter. He pointed out that our airplane
designs evolved fastest in a real "shooting war" and
complained that we weren't enlightened enough to somehow
MOTIVATE OURSELVES to be as creative and resourceful as
we can be WITHOUT bullets zipping past us.
Recently I read a well-researched historical work, "What the
Dormouse Said: How the 60s Counterculture Shaped the Personal
Computer Industry" (book, 2005) by John Markoff,
( http://www.amazon.com/exec/obidos/ASIN/0143036769/hip-20 )
and I was reminded that the PC revolution was fought, especially
in the early days, by people with very specific POLITICAL motivations.
Groups like the People's Computer Company, the Community Bulletin
Board and the Homebrew Computer Club, all on the San Francisco
Peninsula in the 1970s, were coalitions of people who refused to
war work and people who did it eagerly to get the funding for
their radical ideas. And the all wanted to FEED YOUR HEAD, man.
The goals was intelligence amplification, ability augmentation
and individual empowerment.
As much as it sounds like a 60s cop-out, I have found that
"working to change the system from within" is a fruitful
approach. The work I have done post 9/11 for my friend
Dr. Dave Warner and Mindtel
( http://www.mindtel.com )
( http://www.well.com/~abs/HIP/Mindtel/VPHT2.html )
was based on a shared vision of repurposing the military for
humanitarian and public health work.
THIS WORLD IN ARMS
We cannot expect to make everyone our friend, but we can
try to make no one our enemy.
Those who would be our adversaries, we invite to a peaceful
competition -- not in conquering territory or extending
dominion, but in enriching the life of man.
...
Let us build a structure of peace in the world
-- 1st inagural address of Richard Nixon
( http://www.yale.edu/lawweb/avalon/presiden/inaug/nixon1.htm )
quoted in a display at the Nixon Library
( http://www.well.com/~abs/Cyb/4.669211660910299067185320382047/nixon_inaugural.jpg )
( http://en.wikipedia.org/wiki/Nixon_Library )
(I was there to see the Walt Disney backyard train)
( http://en.wikipedia.org/wiki/Carolwood_Pacific_Railroad )
In 1976 Seymour Melman wrote a book I never did read, "The Permanent
War Economy: American Capitalism in Decline,"
( http://www.amazon.com/exec/obidos/ASIN/0671222619/hip-20 )
But the title stayed with me. I realized that was what we are in
danger of becoming once again. We went from a Cold War we couldn't
fight to a War on Terror we can't win.
And I like to recall President Dwight D. Eisenhower's "cross of iron"
speech, as he left office in 1960,
( http://www.eisenhower.utexas.edu/chance.htm )
in which he warned against undue political influence by the
military-industrial complex. The ex-general, who'd won
Europe for us in WWII, also said:
Every gun that is made, every warship launched, every rocket fired
signifies, in the final sense, a theft from those who hunger and
are not fed, those who are cold and are not clothed.
This world in arms is not spending money alone.
It is spending the sweat of its laborers, the genius of its
scientists, the hopes of its children.
The cost of one modern heavy bomber is this: a modern brick
school in more than 30 cities.
It is two electric power plants, each serving a town of 60,000
population.
It is two fine, fully equipped hospitals.
It is some 50 miles of concrete highway.
We pay for a single fighter with a half million bushels of wheat.
We pay for a single destroyer with new homes that could have
housed more than 8,000 people.
This ... is the best way of life to be found on the road the
world has been taking.
This is not a way of life at all, in any true sense. Under
the cloud of threatening war, it is humanity hanging from a
cross of iron.
We have changed course several times since Eisenhower spoke those
words. We have turned the hinge of history. We can do it again.
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