From differential equations to state-space and beyond

We have seen in Chapter 5 how to construct block-diagrams for simulation using integrators and, from there, derive state-space equations. The task was made easier if it was possible to use a chain of integrators, which led to a trivial definition of the state. In many cases that strategy is enough.

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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.

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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.

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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.

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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.

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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.

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