Reinventing Engineering Mathematics

For the 21st Century

What is the Z-Transform of the sequence:

xk = {3,1,4,1,5,9, . . .}

X(z) = 3z0 + z-1 + 4z-2 + z-3 + 5z-4 + 9z-5 + . . .

Hey! X(z)|Z = 10

=

The Purpose of Engineers is to Design Stuff . . .

• Which can be built using well understood technology.

• Which will perform its function(s) in accordance with prescribed specifications.

• Which will not exhibit unexpected, undesirable behaviors (e.g. resonances, instabilities).

The Purpose of Engineering Mathematics is . . .

• To provide tools and skills for analytical and critical thinking, e.g. deduction, induction, abstraction.

• To provide notations and methods to describe and model the Physics of the Natural World.

• To provide tools for modeling and predicting the capabilities of Technologies.

• To provide tools for modeling and predicting the behavior of particular Designs, in interaction with the Natural World.

Underlying Assumptions

• The Natural World is Continuous

• The Natural World is Causal

• The Natural World is Linear . . almost.

• Time does not slow down, speed up, stop, or go backwards.

The Natural World is Continuous

• Quantum and/or molecular effects only rarely enter into Physical Models as discrete functions.

• Differential equations are the primary means for primitive behavioral descriptions of phenomena.

• Differential and integral calculus are essential descriptive tools.

• Continuous mathematical models transcend computation.

The Natural World is Causal

• “Cause and Effect” is the way things work.• Today’s effect is not the result of tomorrow’s

cause.• Simple phenomena: Input/Output Models• Complex systems are modeled by linking simple

models together in block diagrams.

The Natural World is Linear

• A huge class of practical problems yield mathematical models which are linear.

• Many non-linear models can be “linearized”• A large and powerful set of mathematical

tools are available for dealing with linear systems.

• A huge academic investment is made in teaching undergraduate engineers to master these tools.

Time Does Not Slow Down . .

• Newtonian Mechanics is adequate for the vast majority of Engineering Problems

• “Prediction of Behavior” generally implies foretelling the future.

• We need time domain and frequency domain techniques for characterizing and predicting the future behavior of systems.

Enter the Digital Computer• Numerical computation at high throughputs is now

an inexpensive and common component in design solutions.

• The paradigm shift is to discrete behavior.• The relevant mathematical tools recognize this

discreteness: sampling theory, discrete mathematics, theory of computation, finite element analysis, numerical methods, difference equations, and the Z transform.

• Most of these topics are considered “Post-calculus” concepts.

The X-29

• Airframe is designed to be aerodynamically unstable.

• Sample data control using triple redundancy allows stable yet highly maneuverable operation.

• Operating envelope restricted by control system to account for human physiological limits.

Our Pedagogical Dilemma

• More and more emphasis is being placed on complex system analysis and design.

• It takes 3 semesters of Calc plus DE before any rigor can be applied to modeling at the simple component level.

• “Post-Calculus” concepts cannot even be introduced until the Junior year.

• We’re out of time!

Our Dilemma (cont)

• The undergraduate engineering math curriculum does not provide the tools for design in the 21’st century.

• Current curricula and pedagogy prevent these tools from being taught in time to be applied in undergraduate design exercises/projects.

• The basic material in the curriculum is no longer germane to design using current technology (incorporating imbedded computation), other than as foundational material.

...so shoot me!

• Can an engineering mathematics curriculum be designed which is based on computation, difference equations, and the Z transform as foundational material...

• ...and builds the study of calculus, differential equations, and frequency based transforms upon that foundation?

Some Advantages• Less sophistication/maturity required to

grasp behavior of processes in terms of sampled values (sequences) vs. continuous functions.

• No calculus required for Z transforms.

• Strong emphasis on polynomials and complex roots.

• Difficult concepts (e.g. convolution and impulse response) become easy.

Advantages (cont)

• The z-1 (delay) operator is easier to conceptualize than s (differentiation) or 1/s (integration).

• Block diagrams and system analysis can be brought into engineering courses at Sophomore level (maybe Freshman!), rather than Senior.

• Difference equations can be implemented using spreadsheets, lending full visibility to total behavior.

Simple Example: Sine Generator

2*

sin( )

2 cos 1

Kz zH z

z zz p z p

sincos jp

O36.87 1

1 2

0.6 ( )( )

1 1.6 ( )

z Y zH z

z z X z

sinK

Impulse Response

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30

k

y(k)y(k)

Sine

k y(k)0 0.00001 0.60002 0.96003 0.93604 0.53765 -0.07586 -0.65897 -0.97858 -0.90669 -0.4721

10 0.151211 0.714112 0.991313 0.872014 0.403915 -0.225816 -0.765117 -0.998418 -0.832419 -0.333320 0.299021 0.811822 0.999823 0.787924 0.260925 -0.370526 -0.853727 -0.995428 -0.739029 -0.186930 0.4399

Suspected Advantages

• Fundamental understanding of computation.• The ideal foundation for learning calculus

concepts.• Lots of complex arithmetic, algebra, and trig.• Tools and techniques are directly applicable

in current and future application domain.• Early exposure to the Transform concept.

Impacts & Impediments

• Revolutionary change to the way engineering math is taught.

• De-emphasis of continuous mathematics (inevitable consequence of imbedded computers)

• How/when do we teach Physics & Mechanics?

• K-12 (pre-calculus) implications.

• Total overhaul of calculus to leverage new background and speed up delivery.

I & I (cont)

• No textbooks.

• Many math/eng faculty never taught (learned?) Z-transform.

• Inertia: inherent resistance to revolutionary change.

• Risk: What if Jim’s totally wrong?

A Tantalizing Challenge

The Z-Transform has traditionally been considered (wrongly, I believe) to be a post-calculus topic. There is probably no mathematician or engineer who learned calculus after having learned the fundamentals of computation, solutions of difference equations, and the z-transform approach to difference equation solution, system analysis, modeling, and design. We simply don’t know how the concepts of The Calculus would take root in a mind with such a preparation.

It is quite possible that initial comprehension of the fundamentals of calculus would be profoundly improved by such preparation, perhaps even to the extent that what now takes four semesters to teach could be taught in three (perhaps two?), with improved assimilation.

The benefits to undergraduate engineering, especially as we look toward even more pervasive presence of imbedded computing in design and the increasing importance of introducing system analysis techniques earlier in the curriculum, would represent a quantum leap in undergraduate engineering preparation.

What to do?• Plunge in – Total re-write of Engineering/

Engineering Math Curricula starting Fall ’06

• Apply for NSF Curriculum Development Grant (5 year at least)

• Pilot Class – test case

• Footnote in Gen-Ed review process.

• Buy Jim a 6-pack and forget the whole thing . . . . (maybe a 12-pack?)