Question for Tim Wescott

I use root-locus for initial design, when the plant doesn't exist yet and I'm trying to figure out if I need a PID controller, some subset, or something else entirely.

If I'm building a system that needs to have good regulation and disturbance rejection then once I've gotten a chance to get my hands on a built system I generally do a frequency sweep and do design by Bode plot analysis.

The reason I don't use root-locus once I have measured data is that root-locus design requires that you have a transfer function that accurately models the plant, and root-locus design gets difficult when the transfer function order gets high. I find that Bode plot analysis cuts out a number of approximating steps, which is a nice thing.

Actually a unity gain with an open-loop gain of 180 degrees (0 degrees if you include the implied subtraction at the summing junction) means you have a pole on the stability boundary*. The assumption with Bode plot design is that you are starting with a stable system, so your gain and phase margins tell you how much things must change before your poles cross out of the stable region.

• I'd say "unit circle" because I generally do all my design in the same sampled-time domain as my controllers, but the effect is the same in the Laplace domain.

By frequency sweep, i assume you mean like the open loop phase margin measurement that i posted.

This is an interesting site:

Yes. I generally do it for just the plant first, then I design my controller and check its predicted response against a full open loop measurement. I don't think I posted this link before, but here's a description of how I usually do it:

It looks like it. He's incorrect in calling it "sampling" -- it's a cyclically time-varying system for which using the z transform can yield more exact results, but it's not a "sampled" system.

Ok, from Wiki:

"A system is stable if all of its poles are in the left-hand side of the s-plane or inside the unit circle of the z-plane."

So it seems most of my former professors were using the s-plane or Laplace domain, whereas others prefer the z-plane or time-domain.

My control systems class was a long time ago, and my field is mainly RF electronics, so a lot of this pure control theory goes over my head, but i'll ask a simple question anyways:

How would you transform from the s-plane to the z-plane?

My favorite book on the technique...

"Synthesis of Filters" Herrero & Willoner © Prentice-Hall 1966 Library of Congress Catalog Number: 66-27547

...Jim Thompson

There are a variety of methods, each with it's own advantage.

In a closed-loop system I prefer to get an exact transform of a plant's transfer function from the s domain into the z domain by finding it's response to a unit pulse from the controller's DAC and seeing what that looks like at the controller's ADC. This is only exact from the perspective of the controller, however.

For many other systems you can use various approximations. The most popular one is the Tustin, where you use

This works pretty well, and has the advantage that a stable system in the s domain translates into a stable system in the z domain. It has a disadvantage that the locations of the poles crawl around, but they do so in a predictable way that you can overcome by 'prewarping' the poles and zeros. This is pretty good going both ways, so it is a way to take a continuous-time analog filter and translate it into a discrete-time filter.

You can also do the whole thing by frequency-domain analysis, where you calculate the attenuation through the entire channel, taking your anti-alias filter, any DSP filtering, and the DAC zero-order-hold effect into account. Those few times when I need to do this task, this is how I accomplish it.

Ah yes, another wiki page that soars over my head:

But it's 1 + sT/2 z ~ -------- . 1 - sT/2

I immediately recognized the bilinear transformation for the Smith Chart, which transforms from complex impedance (or admittance) to a unity circle reflection coefficient (Gamma), which includes magnitude and angle: