I'm a faculty member in Mechanical Engineering at Penn State and
I've been 'listening in for about a year. I specialize in robotics and
applied control systems, usually position control and have recently
developed a graduate level class in neural networks applied to
control systems. I thought I'd try to respond to Gary's request for a
description of Laplace transforms, so here it goes:
Laplace Transforms in 100 words or (more or) less:
The Laplace Transform is a mathematical technique used in the
solution and analysis of ordinary (and sometimes partial)
differential equations. You might think of it as analogous to
logarithms which can transform multiplicative operations to
additive ones (the principle upon which the slide rule is designed).
Laplace Transforms can transform differential equations into
algebraic ones.
(That's 55 words so far...)
Laplace transforms are an extremely useful tool in the analysis and
design of feedback control systems. They led to the development of
several powerful graphical tools (Bode plots, Nichols Charts,
Nyquist diagrams) which were essential to the design of feedback
control systems in the post WW II era (before computers became
widely available).
(Oops, I'm over my limit and I still haven't answered Gary's question,
please bear with me for another paragraph...)
Why would it be useful for the PCT crowd? The main advantage of
the Laplace transform is that it allows us to consider dynamic
systems in the 'frequency domain', as opposed to the time domain,
which is where you are now. Concepts like time constants and 'leaky
integrators' (which I saw pop up the other day) have easily grasped
and conveyed parallels in the frequency domain. They translate to
'system bandwidths' and corner frequencies. In particular there are
some very powerful stability criteria available to analyze systems
with feedback loops. This might be of use.
One final caveat. The Laplace Transform (and hence, all of the
above-mentioned tools) work only on LINEAR systems. So far, all
I've seen in the discussions of simulations and models are linear.
But as soon as you introduce products of time varying signals, or
nonlinear functions (like sine of an elbow angle....) then the tool is
pretty much worthless unless you want to linearize the problem
about a specific location in the variable space, in which case, the
analysis becomes valid for only small excursions away from that
operating point.
Well, it's a bit more the 100 words, but I tried to keep it brief. I
hope I've been helpful. I'll go back into the woodwork now....
John F. Gardner
Asst. Professor of Mechanical Engineering
Penn State
University Park, PA 16802
(814) 865-8281