Theory – Page 2 – Fundamentals of Linear Control: A Concise Approach

Balance scales and integral action. Part II

In a previous post we discussed how the principles of operation of a balance scale could be understood as a feedback loop with integral action. We left off after analyzing the impact of measurement disturbances in the loop and mentioned that input disturbances could be used to model “operator errors.”

Balance scales and integral action. Part I

You can find many accounts of the history of feedback control in books, articles, and online. Some will go back to Greek and Arab antiquity to identify proportional feedback in water clocks, wind mills, advancing to the industrial revolution and the Watt governor. In virtually all of those accounts the feedback loop involves continuous-time signals. So I find it interesting that there is device that man has been using for millennia that requires a discrete-time feedback algorithm and integral action for its operation. That device is a balance scale.

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!

Is a proportional controller right for you? Ask your control engineer! Part III

In Part I and Part II of this post we looked into the problem of finding a suitable discrete-time controller when the system being controlled is fast to reach steady-state so that it can be modeled as the single delay

$$y(t) = G u(t – \tau), \quad \tau > 0.$$

In Part I we showed that a proportional controller is not a good choice and in Part II we found out that, surprisingly, an integral only controller is an excellent choice but requires the use of a gain $\hat{G}$ that has to match the gain of the system $G$. In the present post we will revisit the design of a dynamic controller with the goal of showing that:

the Smith predictor from Part II also had a zero;
the integral controller is indeed an excellent choice;
the mismatch $\hat{G} \neq G$ is not necessarily catastrophic.

Is a proportional controller right for you? Ask your control engineer! Part II

Let’s now take another look at the problem we introduced here in this post. The problem was that of finding a suitable discrete-time controller when the system being controlled is fast to reach steady-state so that it can be modeled as the single delay

$$y(t) = G u(t – \tau), \quad \tau > 0.$$

As seen before, a proportional control is not a good solution in this case. So what is the “simplest” controller one could think of in a situation like that? Of course it will be a dynamic controller! We will go about constructing one such dynamic controller in the rest of this post.

Is that a constant? Or is it a delta?

This might be a stupid question.. but oh well. So the inverse laplace of a constant is the dirac delta function times the constant.
With a proportional controller, K(s) = Kp, the inverse laplace of the controller would be the delta function. Is the delta function asymptotically stable?

Anonymous

As it turns out, this is not a stupid question at all! It hits right at the heart of what is and how to represent a dynamic system.

Is a proportional controller right for you? Ask your control engineer! Part I

In another undergraduate class I teach here at UCSD students develop hands on control projects and many of them have not taken a class in controls. They have to quickly design and implement controllers often using really cheap and noisy sensors and slow processing times. In that context, a proportional controller, which might be the first choice in a search for a suitable controller, is rarely adequate. I will deal with the issue of noisy sensors and how they severely limit the closed-loop bandwidth in another post. In this one I want to address the issue of large delays.