Page 75 - Modern Control Systems

C H A P T E R Mathematical Models
2 of Systems

2.1 Introduction 50
2.2 Differential Equations of Physical Systems 50
2.3 Linear Approximations of Physical Systems 55
2.4 The Laplace Transform 58
2.5 The Transfer Function of Linear Systems 65
2.6 Block Diagram Models 79
2.7 Signal-Flow Graph Models 84
2.8 Design Examples 90
2.9 The Simulation of Systems Using Control Design Software 113
2.10 Sequential Design Example: Disk Drive Read System 128
2.11 Summary 130

PREVIEW

Mathematical models of physical systems are key elements in the design and analysis
of control systems. The dynamic behavior is generally described by ordinary differen-
tial equations. We will consider a wide range of systems, including mechanical,
hydraulic, and electrical. Since most physical systems are nonlinear, we will discuss lin-
earization approximations, which allow us to use Laplace transform methods. We will
then proceed to obtain the input-output relationship for components and subsystems
in the form of transfer functions. The transfer function blocks can be organized into
block diagrams or signal-flow graphs to graphically depict the interconnections. Block
diagrams (and signal-flow graphs) are very convenient and natural tools for designing
and analyzing complicated control systems. We conclude the chapter by developing
transfer function models for the various components of the Sequential Design
Example: Disk Drive Read System.

DESIRED OUTCOMES
Upon completion of Chapter 2, students should:

U Recognize that differential equations can describe the dynamic behavior of physical
systems.
U Be able to utilize linearization approximations through the use of Taylor series
expansions.
• Understand the application of Laplace transforms and their role in obtaining transfer
functions.
• Be aware of block diagrams (and signal-flow graphs) and their role in analyzing
control systems.
3 Understand the important role of modeling in the control system design process.

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