Theory – Fundamentals of Linear Control: A Concise Approach

Z-transform and rational functions as a database for sequences

The Z-transform is essentially a clever database that can store and operate on the values of a discrete-time signal. The inverse Z-transform can be used to query the database.

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.

More on the Fourier transform

In a previous post we explored some relationships between the Laplace and the Fourier transforms. In this post we address a common question: “when is the Fourier transform of a signal, $X(j\omega)$, equal to its Laplace transform, $X(s)$, evaluated at $s = j \omega$?” The short answer is: when the region of convergence of the corresponding Laplace transform contains the imaginary axis.

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.

The impulse response

So you wonder how impulse responses are born? They are born out of linearity.

The “correct” phase of the frequency response

When plotting the phase of a model’s frequency response there will always be ambiguity on representing angles, after all angles are equivalent modulus $2 \pi$. However, when plotting the frequency response in a Bode plot, it is customary to unwrap the phase and display the plot as a curve whose image often spans more than $2 \pi$. In that sense, one should also care about any potential integer multiples of $2 \pi$ shifts of the curve. Yet, I have never seen a discussion on what the “correct” shift should be.

Asymptotic tracking a constant reference without a pole at zero?

On a previous post we have discussed how by invoking stability one can solve problems without having to worry about particular inputs. The contrast between a solution that holds for a particular input versus a solution that holds for a class of inputs is highlighted in the present post by revisiting the problem of asymptotic tracking for a simple example.