I'm looking for a good textbook or website or course or other
reference that has a better motivated approach to the Laplace
transform. Most authors simply define the transform, then show how
useful it is, leaving the student to scratch his or her head as to
what the transform means or where it comes from. I'm looking for
something that starts from the idea that the complex exponentials are
the eigenfunctions of LTI systems, and that the Laplace Transforms
(and Fourier Transforms, for that matter) are analogous to taking an
infinite dimensional dot product. Is there anything out there that
emphasizes this approach?
I always start with Wikipedia and Wolfram Mathworld on the web. The
Laplace transform is treated in lurid detail in most control texts.
Unfortunately, you don't get a lot of intuition about it before your
head swims. Here is my stab at intuition (shoot me if I got the
mathematical details wrong):
If you let s=jw (w=omega - frequency in radians) e^-st = cos (wt)-j sin
(wt) [Euler's theorem]. This describes a unit vector that rotates
about the origin in the complex plane. When you integrate your time
function by multiplying by this function, it composes your function
into an algebraic equation that represents your function over
frequency. In a sense it is saying for each frequency, lets figure out
the total sum of the function's contribution to that frequency by
adding it up over time zero to infinity.
The salient point of linear systems is that their output can be
composed of a sum of all the inputs independently. That is the
critical difference with non-linear systems - where this does not
So... a sinusoidal input at each frequency will result in a response
that has a new amplitude and phase relative to the input. This is
repeated over the whole frequency spectrum. If you decompose a step
function or any other input to the system in the same way, you "make"
the input into a collection of such sinusoids. For example, a unit
step is 1/s. One important point: the Laplace transform begins at
time=0. So it is based on a signal that begins at or after 0. (The
Fourier transform is similar, and for periodic signals that are
assumed to exist from -infinity to infinity.)
A common way to look at the function in the frequency domain is the
Bode plot. It shows magnitude in db, and phase in degrees. There are
many others such as Nyquist.
In the frequency domain, there are a number of useful properties that
give you gain, starting value to an input, final value, etc. in the
time domain. You can also infer stability,etc. I encourage you to
look these up.
Mr. Laplace had the genius to observe that once you have this
transformation, the task of solving differential equations becomes an
algebraic exercise. In the old days you would do some algebra to
separate out the terms and then look up the inverse transforms.
Finally you would calculate the time response as a sum.
In modern days we're too lazy to do this, ( unless we're in a controls
class). We can use a simulation program to help us do the analysis.
But... we need to know how to interpret the results.
Dynamic Simulation Made Easy
Thanks for the recommendations, but the answers were not quite what I
was looking for. I think I didn't ask the question correctly. Let me
I'm considering writing a hypertext control / linear systems /
dynamics "text". As you know, the subject is fairly involved and
depends on a significant volume of mathematics: differential
equations, complex variables, linear algebra, transform / distribution
theory, and physics. There's really no obvious way to introduce all
these subjects to the beginning student (unlike, say calculus, which
has a pretty standard pedagogical hierarchy of e.g. functions, limits,
derivatives, integrals). As a result, most texts simply introduce the
Laplace transform out of nowhere, then try to convince the student how
useful it is. My idea is to better motivate the subject by introducing
the concept that complex exponentials are the eigenfunctions of LTI
systems, along with the analogy between signals and (infinite
dimensional) vectors (after discussing what eigenvectors and
eigenvalues are for matricies). Hence I think the Laplace transform is
nothing more than the dot product of a signal (or system) with the
complex exponential basis function. The Shaum's outline book of
Signals and Systems has some of this view. Siebert's Circuits Signals
and Systems (my prof way back when) also has some of this flavor in an
appendix, but neither book quite goes into sufficient detail. (BTW I
think the hyper linked format will work nicely to allow the student to
bring in the math as necessary.) Anyway, I'm looking for a treatment
with this "vector space" emphasis. I think, if done correctly, that
this approach might have a more intuitive feel.
Here's an example of the kind of thing I'm going for. Have you noticed
that most linear algebra (and controls) texts define the determinant
of a matrix by the procedure used to compute it, without ever really
mentioning what it IS. If you view the matrix equation y = Ax as an n-
dimensional transformation from the "x" space to the "y" space, the
determinant is just the ratio of n-dimensional volume or area between
the two spaces. That's my concept - to provide an intuitive (if not
necessarily mathematically rigorous) explanation of what things "are",
rather than just how to calculate them.
If you can do it, and end up with something that is comprehensible to
the average undergraduate, then more power to you.
The textbook that I have that comes closest to that is Harry VanTrees
"Detection and Estimation Theory". It doesn't go into the Laplace
transform as you mention, but it provides all the foundation that I need
to immediately see where you're coming from. It is also the text from
the single most difficult class that I have ever taken (when the prof
tells a dozen really bright grad students "you'll pull together and I'll
give 12 A's, or you won't and I'll give 12 D's", and ends up being
right, you know it's a hard class).
So I can certainly see the value of it, if you make it fly. (In fact, I
want to read it, if you can make it fly).
Something that you may want to consider as you launch into this: my
introduction to control theory came as part of a program that put
control theory _after_ you had taken a comprehensive course in signals
and systems (good ol' Oppenheimer, Willsky & Young). So while the text
introduced the Laplace transform, we just skipped that part. I suspect
that most controls texts are taught in that sort of environment, where
the student is assumed to have been introduced to the LT already, and it
is only included in the book in case the student parachuted into the
program from a different planet.
So you may go to all that effort, and have it ignored anyway.
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