6_System_Def - Chapter 6 Introduction to Systems Chapter...

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105 Chapter 6 Introduction to Systems Chapter Outline 6.1 DEFINITION OF A SYSTEM. ............................................................... 107 6.1.1 Definitions. ................................................................................................................................... 107 6.1.2 Representations of Systems . ................................................................................................ 110 6.2 SYSTEM REPRESENTATIONS. ........................................................... 111 6.2.1 Introduction to System Representations. ........................................................................ 111 6.2.2 Differential Equations . ............................................................................................................ 112 6.2.3 Laplace Transfer Functions. ................................................................................................. 112 6.2.4 Convolution Integral . ............................................................................................................... 113 6.2.5 Fourier Transfer Function. ..................................................................................................... 114 6.2.6 Response to Standard Inputs. ............................................................................................... 114 6.3 ELECTRICAL NETWORKS . ................................................................ 116 6.3.1 System Representations. ........................................................................................................ 116 6.3.2 The Step and Impulse Responses. ..................................................................................... 119 6.4 MASS-SPRING-DAMPER SYSTEM. ..................................................... 120 6.4.1 System Representations. 120 6.4.2 The Step and Impulse Responses. 123 6.5 PROOF-MASS ACTUATORS . 124 6.5.1 Linear Motors as Actuators . 124 6.5.2 System Representations. 126 6.5.3 The Step and Impulse Responses. 127 6.6 CHAPTER SUMMARY. 128 6.6.1 Definitions. 128 6.6.2 Comparison of Impulse and Step Responses for the Three Systems. ............... 129 6.7 HOMEWORK FOR CHAPTER 6. .......................................................... 130 In Chapter 5 we introduced the concept of a signal. This concept is one of the most fundamental concepts in system theory. In this chapter we use the concept of a signal to define the second fundamental concept of system theory: a system. As described in Chapter 1, the input signal to a system produces an output signal. This definition is very general, and it allows us great flexibility in applying it to a variety of physical processes. At the same time there are a number of very powerful system theoretic tools available that we can use to analyze very complex systems. It is the purpose of this text to begin the development of the concepts and tools related to systems. A system is given concrete mathematical expression by an explicit mathematical description of the system called a model . The premise of system
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106 Chapter 6 Introduction to Systems theory is that a mathematical model can be used to gain insight into a physical processes. Implied in this statement is that: (1) we can find a model that adequately describes the physical process, (2) this model is suitable for analysis, and (3) this model is suitable for design (if applicable). It is important to recognize that a model can accurately describe the physical observations, but be too complicated for analysis or design. Or a model that lends itself to analysis and design can be too simplistic to accurately describe the observations. In order to choose an appropriate model, it is important to understand what system analysis techniques exist, and to what kinds of system models these analysis techniques apply. These analysis techniques include both mathematical tools and computer-aided design packages. It is the purpose of this text to develop this background. It is the purpose of this chapter to introduce several different kinds of models, called representations, that we will study in the coming chapters. In order to firmly fix the concept of a system, we will also introduce several systems that we will use to illustrate the concepts in the coming chapters.
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6_System_Def - Chapter 6 Introduction to Systems Chapter...

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