May 2019 – Page 2 – Fundamentals of Linear Control: A Concise Approach

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.

Simulating linear systems with non-zero initial conditions

I have discussed in another post how to work with state-space models to simulate the response of linear systems using Matlab’s lsim. It is however also possible to approach this problem from a transfer-function point of view without having to mess with state-space at all.

Simulating linear systems with non-zero initial conditions in state space

Matlab’s lsim function for simulating linear systems will give you the option to provide an initial condition if your system is in state-space but not for transfer-functions.

Simulating linear systems with Matlab’s lsim

In this post we will go over a couple of ways to simulate linear systems in Matlab.

Routh-Hurwitz stability criterion

I am often asked why I chose not to cover the Routh-Hurwitz stability criterion in the book. Here are some of the reasons.

First is space. I would need to develop a lot of ideas to be able to do a decent presentation of the Routh-Hurwitz criterion. That takes space.
Second is lack of connection with other ideas in the book. The algebraic polynomial machinery that is behind the criterion is somewhat disconnected from the other topics covered.
Third is utilitarian. For system with numeric coefficients the use of the criterion has been virtually made obsolete by numeric calculators. Finding the roots of high order polynomials in the 21st century is not as hard as it used to be in 1876! For symbolic coefficients, the algebraic rational inequalities produced by the application of the criterion are only useful for relatively small problems. They do not scale very well or are as universal as the root-locus method or the Nyquist criterion. In the end one still has to solve a set of rational inequalities in the system’s coefficients to assess stability.
Fourth is the Pandora’s box effect. If I cover Routh-Hurwtiz, why not the Liénard–Chipart criterion? Or the Hermite criterion? Those are even nicer than the Routh-Hurwitz one!

What do you think?

New slides

I have posted new and updated slides. The new slides are based on reveal.js and they have two dimensions: you move up and down on a topic and sideways between sections. Try hitting <ESC> to have an overview of the content. Let me know if you find any mistakes or typos.