# 11_SS - Chapter 11 Introduction to Realization Theory...

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355 Chapter 11 Introduction to Realization Theory Chapter Outline 11.1 CALCULATION OF A TRANSFER FUNCTION FROM A STATE SPACE REPRESENTATION. .............................................................. 357 11.1.1 Introduction. ............................................................................................................................... 357 11.1.2 State Space Representation to Transfer Functions. ............................................... 357 11.1.3 Basic Relationships Between the Transfer Function and State Space Representations . ..................................................................................................................... 361 11.1.4 MATLAB Experiments. ....................................................................................................... 364 11.2 TWO REALIZATIONS. ....................................................................... 365 11.2.1 Introduction. 365 11.2.2 First Realization. .................................................................................................................... 366 11.2.3 Second Realization . .............................................................................................................. 371 11.2.4 MATLAB Experiments. 372 11.3 EQUIVALENT DYNAMICAL SYSTEMS. ............................................ 373 11.3.1 Introduction. 373 11.3.2 Transformations of States . .................................................................................................. 374 11.3.3 Input-Output Relationships. ................................................................................................ 376 11.3.4 MATLAB Experiments. 377 11.4 STATE EQUATIONS FROM PHYSICAL LAWS. ................................. 378 11.4.1 A Network Example . ............................................................................................................. 378 11.4.2 Phase Variables. ...................................................................................................................... 380 11.4.3 Incorporation of Initial Conditions into State Space Equations . ...................... 382 11.4.4 MATLAB Experiments. 385 11.5 MULTIVARIABLE SYSTEMS . ........................................................... 386 11.5.1 Introduction. 386 11.5.2 State Space Representations . ........................................................................................... 386 11.5.3 Transfer Functions. ................................................................................................................. 390 11.5.4 MATLAB Experiments. 391 11.6 CHAPTER SUMMARY. 393 11.7 HOMEWORK FOR CHAPTER 11. ...................................................... 394 In the last chapter we introduced the state space representation as an outgrowth from an all-integrator block diagram. It is obvious that there is a close relationship between the transfer functions, block diagrams, and the state space representations. In this chapter we will develop many of these relationships. In general, the theory that describes this relationship between transfer functions and state space representations is called realization theory; hence, the chapter title.

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356 Chapter 11 Introduction to Realization Theory There are three themes that are developed in this chapter. The first theme is the relationship between a state space representation of a system and its transfer function. First, we give the formula for computing the transfer function from a state space representation. This formula is straightforward. We also note several key relationships between the state space system and its transfer function. Second, we develop a method for translating a transfer function into a state space representation. This reverse direction, developing a state space representation from a transfer function, is difficult. Most of the theory is beyond the scope of this text. We present two simple ways of translating a transfer function or block diagram into a state space representation, which are sufficient for our purposes. Third, we develop the relationship between state space representations that have the same transfer function. This result indicates the flexibility of the state space representation for modeling and analyzing a system. The second theme that is discussed in this chapter is the development of a state space representation directly from the differential equations of the system that are derived from physical laws. We also discuss how to translate a differential equation into a state space representation for systems which are governed by higher order differential equations. When these insights are combined with the previous results in this chapter, the state space representation emerges as an extremely powerful tool for modeling and analysis of systems.
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## This note was uploaded on 09/10/2009 for the course ECE 60367 taught by Professor Meehan during the Spring '09 term at Virginia Tech.

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11_SS - Chapter 11 Introduction to Realization Theory...

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