## Happenings – 2012 Sep 22

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.

Posted in diary. 2 Comments »

### 2 Responses to “Happenings – 2012 Sep 22”

1. rip Says:

OK, I’ve got it. I’m disappointed at how few books say anything about this. After I figured it out, I saw that even “Feedback Control of Dynamic Systems” by Franklin, Powell & Amami-Naeini just barely touches on it, but enough to refresh my vocabulary.

The Bode plot deals with the forced response – the particular solution – of the system to a sinusoidal input; that forced response is of the same frequency, but the amplitude and relative phase of the output are functions of the frequency (for linear time-invariant systems).

The Bode plot says nothing about the natural response of the system – that is, about the solution of the homogeneous equation. If the natural response decays, then the forced response is the long-term output, and we’re fine… but if the natural response does not decay (repeat after me: a pole in the right-half-plane), then it is the natural response, rather than the forced response, that is the long-term output.

So, my Bode plot correctly says that the forced response goes to zero very quickly… unfortunately, the natural response of this system explodes, and the Bode plot is silent about the natural response.

• rip Says:

Oh, actually the Bode plot does say something about the natural response: that peak at a frequency of .3 rad/sec identifies the natural frequency – because if we force the system at its natural frequency, then we get resonance or partial resonance.