Chapter 6 – Fundamentals of Linear Control: A Concise Approach

The Fourier and the Laplace transforms

Because feedback requires careful consideration of stability and causality, the transform tool of choice is almost always the Laplace transform. Yet, in many applications it is the Fourier transform that arises most naturally.

Let your inputs go!

As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the book. Many students, when asked what they remember from their undergraduate control class, will quickly point at the root-locus or at the Nyquist stability criterion. Yet, in the beginning, it might seem that stability is a side dish and not the main attraction.

Controllers gone unstable

In order to obtain asymptotic tracking, we need $S(0)=0$, which means we’d need one of the poles of $G K$ to be zero. If $G$ has no poles at zero, then $K$ must be the transfer-function to have the pole at zero. But in order to be asymptotically stable, we need the real parts of the poles to be less than zero, correct? So does it matter if our controller is not asymptotically stable? Is there a way to design a controller that is both asymptotically stable AND provides asymptotic tracking?
Anonymous

Root-locus angle of departure and arrival

I have previously discussed about why I kept the number of root-locus “rules” at a minimum in this post. One popular “rule” that I have omitted is the calculation of departure or arrival angles at complex poles or zeros.

Root-locus breakaway/breakin points

As I mention in Chapter 6, one can provided additional root-locus “rules” that can help refine plots done by hand. My approach when writing the book was to keep the number of rules to a minimum, reflecting the fact that one will rarely draw a root-locus diagram by hand these days. In my opinion, the main goal here should be to learn how adding, removing, or moving poles and zeros can impact the overall root-locus. Not how to accurately sketch the root-locus. That goal can be accomplished comfortably with the limited set of rules provided in Section 6.4. Yet, every now and then I will get a question such as the one below:

I looked through the book, and I probably missed it, but if we have a root locus that diverges from the real axis and goes complex how do we determine the point at which it diverges? I get that for some plots it will diverge onto the imaginary axis at our center of asymptotes, but for some more complicated plots they diverge from a different point.
Anonymous

Yes, there’s a rule for that!