Chapter 3. Dynamic Behavior

The goal of this chapter is to understand dynamic behavior. We begin by working with linear state space models, often obtained by linearizing a nonlinear model, such as those developed in Chapter 2. We then introduce Laplace transforms. The main advantage to Laplace transforms is that they allow us to analyze behavior exhibited by linear differential equations by using simple algebraic manipulations. Laplace transforms are used to create transfer function models, which are the basis for many control system design techniques.

After studying this chapter, the reader should be able to:

• Apply the initial and final value theorems of Laplace transforms

• Understand first-order, first-order + dead time and integrating system step responses

• Understand second-order under-damped behavior

• Understand the effect of pole and zero values on step responses

• Convert state space models to transfer functions

The major sections of this chapter are as follows:

3.1 Background

3.2 Linear State Space Models

3.3 Introduction to Laplace Transforms

3.4 Transfer Functions

3.5 First-Order Behavior

3.6 Integrating System

3.7 Second-Order Behavior

3.8 Lead-Lag Behavior

3.9 Poles and Zeros

3.10 Processes with Dead Time

3.11 Padé Approximation for Dead Time

3.12 Converting State Space Models to Transfer Functions

3.13 MATLAB and SIMULINK

3.14 Summary