Respect the Unstable

Back in August of 2003 I read an article in IEEE Control Systems Magazine titled “Respect the Unstable”, pp. 13-25, by Gunter Stein. The article made an immediate impression on me. The thrust of the article falls into what I call the “Controllers are Not Magic” category.

Most of the engineers I’ve worked with do not fully appreciate what controllers actually are. Nor do they fully understand what a controller can and cannot accomplish. So every so often I get requests for “Magic” in the form of “Don’t worry about Problem A, we can fix that with the controller.” Let me restate the obvious for anyone new to the controls field – Controllers are NOT Magic they manipulate physical systems and as such acquire the limitations of those systems.

The article by Dr. Stein provides an interpretation of the Bode Integrals as well as several examples of unstable systems.

The Bode Integrals

I’d put the equations here but I haven’t figured out how to do that in WordPress yet. So I’ve included a link the Bode Integrals on the wiki site here:
Bode Integrals in the Wiki

Here are my highlights from the article:

Basic Facts of Unstable Plants

Unstable systems are fundamentally, and quantifiably, more difficult to control than stable ones.

Controllers for unstable systems are operationally critical.

Closed-loop systems with unstable components are only locally stable.

and

The first integral applies to stable plants and the second to unstable plants. They are valid for every stablizing controller, assuming only that both plant and controller have finite bandwidths. In words, the integrals state that the log of the magnitude of sensitivity functionof a SISO feedback system, integrated over frequency, is constant. The constant is zero for stable plants, and it is positive for unstable ones. It becomes larger as the number of unstable poles increases and/or as the poles move farther into the right-half plane. (Technically, we must count all unstable poles here, including those in the compensator, if any.)

This is the equivalent of a conservation of energy for systems under control. The result being that to make a system less sensitive to disturbances in one frequency range the controlled system must become more sensitive in another frequency range (or all other frequency ranges).

In my current position we are often designing controllers to have good disturbance rejection at low frequencies while shooting for a minimum of peaking. The Bode Integrals make it obvious that low frequency disturbance rejection means more peaking in the higher frequencies. Sometimes the largest magnitude of the peaking can be reduced but that just means the peaking gets spread out in frequency. This observation of the Bode Integrals informs us of the limitations of controllers – we can’t have it all. So we now realize that our controller design is largely a matter of choosing the shape of our frequency response very carefully.