An Introduction to System Modeling and Control

2Laplace Transforms

In Chapter 01 we gave examples of physical systems modeled by differential equations. For each such physical system we will use the differential equation model to design algorithms to control the system. Furthermore, it turns out that these algorithms will also be described by differential equations. To ease the difficulty of working with differential equations, classical control uses the Laplace transform. This transform converts a differential equation to an algebraic equation greatly simplifying mathematical calculations. In Chapters 9 and 10 we will present specific approaches to the design of feedback controllers for physical systems based on their Laplace transform model. All of this requires a thorough background knowledge of Laplace transforms which we cover in this chapter.

Definition 1 Laplace Transform

Let for denote a time function. Define the Laplace transform of as

(2.1)

where is a complex number.

Notation The symbol “” means by definition ...