LEARNING OBJECTIVES FOR THIS CHAPTER

1. 15–1 To use the finite-difference approximation of a derivative to develop an approximate discrete-time model corresponding to a continuous input–output model.

2. 15–2 To derive discrete-time state models for linear dynamic systems.

3. 15–3 To develop block diagrams of a digital control system including sampling and holding devices.

4. 15–4 To use the z transform to develop pulse transfer functions of discrete-time systems.

INTRODUCTION

In almost all existing engineering systems, the system variables (input, output, state) are continuous functions of time. The first 14 chapters of this book deal with this category of systems, classified in Chap. 1 as continuous dynamic systems. The last two chapters are devoted to discrete-time systems in which, according to the definition given in Chap. 1, the system variables are defined only at distinct instants of time. It may seem that there are not many such systems, and, indeed, very few examples of intrinsically discrete engineering systems come to mind. There are, however, many systems involving continuous subsystems that are classified as discrete because of the discrete-time elements used to monitor and control the continuous processes. Any system in which a continuous process is measured and/or controlled by a digital computer is considered discrete. Although some variables in such systems are continuous functions of time, they are known only at distinct instants of time determined by the computer sampling frequency, and therefore they are treated as discrete-time variables.