discrete-time – Fundamentals of Linear Control: A Concise Approach

Sampling. Part III

In previous posts we have explored sampling of continuous-time signals and introduced the Z-transform as a tool to work with discrete-time signals. In this post we address the other end of the process, that is the reconstruction of a signal from its samples.

Discrete-time systems and the Z-transform. Part I

The Z-transform does to discrete-time signals and systems what the Laplace transform does to a continuous-time signals and systems.

Sampling. Part II

In an earlier post we discussed how to obtain the continuous-time Laplace and Fourier transforms of a sampled signal based on the Laplace and Fourier transforms of the original signal. We shall now explore other relationships between those transforms and the Z-transform of the sampled signal.

Sampling. Part I

Modern control systems are typically implemented in computers that work with periodic samples of signals, that is discrete-time signals, rather than continuous-time signals. A system that produces samples of a given signal is a linear system, albeit a time-varying one. In this post, which starts a series exploring the issue of sampling and discretization, we delve into the nature of a system that can produce such signal samples. More will follow.

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 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.