TABLE OF CONTENTS Most interesting plants are what can be termed "fully analog." A DC motor, for example, takes a continuous-time voltage as its input and produces in return a shaft-angle also a continuous-time signal. Controllers, however, are another story. Today control algorithms are often implemented by microcontrollers or microprocessors. These devices process their input at fixed times and change their output at fixed times.
We have learned how to describe systems whose input and output change continuously. For this task we use the Laplace transform and transfer functions. Now we are going to learn how to model systems that process discrete-time signals, and we are going to learn how to models hybrid systems systems that have some parts that work in continuous time (like a motor) and some parts that work in discrete-time (like a microcontroller). The z-transform is the correct tool for modeling many discrete-time systems, so we start by defining it and describing some of its properties. We also give a fairly complete description of the bilinear transform and its uses in the control of hybrid systems. Next, we consider the more general problem of designing controllers for hybrid systems. Finally we give a brief description of the modified z-transform and its uses.
