In one respect it’s been another very slow week: I’ve had a cold since dawn last Sunday… and when the congestion was bad enough to make my eyes hurt, I bailed on mathematics. I watched lectures from The Teaching Company on 19th century European history.
I had made some progress on a first control theory post, but that stopped fairly soon… hence, no post last Monday. Ironically, to my delight, the blog got more than 300 hits a day Monday thru Thursday.
Right now, nevertheless, I have a controls post thru stage IV: mathematics and commentary are complete. I just need to move text and images to the blog this wekend, and do final edits as usual Monday evening.
Oh, if it were only that simple!
I kept working examples, and I’m already in trouble. I have a Bode plot – two of them even – which I believe says that the output from a unit-amplitude sinusoidal input should be substantially less than 1 for any frequency greater than about .45 rad/sec.
But the actual solution grows, instead of shrinking. Here’s what I get for the time-domain response to a frequency of 1, i.e. to sin(t).
And both Mathematica version 8 and Control System Professional in version 7 agree: the same Bode plot, and the same explosive time-domain behavior. See for yourself. Here’s Control System Professional…
… and here’s version 8:
Both of those Bode plots say that the output is of very small magnitude for higher frequencies. But that output at freq = 1 is anything but small.
What’s going on?
Damned if I know. And I’m not sure I have the nerve to put out the first post when my very next work seems self-contradictory.
We’ll see what happens. Maybe I’m supposed to work instead on number-theoretic functions this weekend.