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Chapter 11
Introduction to Realization Theory
Chapter Outline
11.1 CALCULATION OF A TRANSFER FUNCTION FROM A STATE
SPACE REPRESENTATION.
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11.1.1 Introduction.
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11.1.2 State Space Representation to Transfer Functions.
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11.1.3 Basic Relationships Between the Transfer Function and State Space
Representations .
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11.1.4 MATLAB Experiments.
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11.2 TWO REALIZATIONS.
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11.2.1 Introduction.
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11.2.2 First Realization.
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11.2.3 Second Realization .
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11.2.4 MATLAB Experiments.
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11.3 EQUIVALENT DYNAMICAL SYSTEMS.
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11.3.1 Introduction.
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11.3.2 Transformations of States .
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11.3.3 InputOutput Relationships.
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11.3.4 MATLAB Experiments.
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11.4 STATE EQUATIONS FROM PHYSICAL LAWS.
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11.4.1 A Network Example .
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11.4.2 Phase Variables.
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11.4.3 Incorporation of Initial Conditions into State Space Equations .
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11.4.4 MATLAB Experiments.
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11.5 MULTIVARIABLE SYSTEMS .
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11.5.1 Introduction.
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11.5.2 State Space Representations .
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11.5.3 Transfer Functions.
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11.5.4 MATLAB Experiments.
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11.6 CHAPTER SUMMARY.
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11.7 HOMEWORK FOR CHAPTER 11.
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In the last chapter we introduced the state space representation as an outgrowth from
an allintegrator 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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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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 Spring '09
 MEEHAN

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