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7/24/2019 60237 Chapter 6 Notes
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Chapter 6
State Variable Analysis
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Let us define
Then the nth order differential equation is decomposed into n first order differential
equations.
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The state Space model can be derived from the state equation and output equation
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Transfer function Decomposition Techniques:
The process of obtaining state model from the transfer functionis called Transfer function Decomposition.
Example:
Let us consider the following transfer function in Eqn (1).
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The transfer function can be decomposed and converted into statemodel using three different types of decomposition techniques namely,
(i) Direct decomposition technique
(ii)
Cascade decomposition technique
(iii) Parallel decomposition technique
(1) Direct Decomposition Technique:
In direct decomposition technique all the terms in the numerator aremultiplied to a single polynomial and all the terms in the denominatorare multiplied in to single polynomial. If the order of the transfer
function is N, then N number of integrators and N number ofsumming points are required to represent in block diagram.
For example consider the transfer function in Eqn (3),
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Example 2:
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The State equation is
The State model is
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2. Cascade Decomposition Technique:
The transfer function is rearranged in cascade (series) form such way that
the decomposed terms are to be multiplied to get the original transfer
function.
Example :
Consider the transfer function given in Eqn (6).
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3. Parallel Decomposition Technique:
The transfer function is rearranged in parallel form such way that the
decomposed terms are to be added (using Partial fraction Expansion
method.) in order to get the original transfer function.
Example:
Consider the transfer function in Eqn (7).
)4)(3)(2(
1
)(
)(
+++
=
ssssU
sY . (7)
Using Partial fraction expansion form, Eqn (7) can be written as
)4()3()2()(
)(
+++=
s
C
s
B
s
A
sU
sY
The values are A, B, and C are computed and now the transfer function is,
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The output of each integrator is assigned as a state variable as shown in
Fig 6. Now the state equation and out put equations are obtained from Fig 6.
The State equations:
The State Model is
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Solution:
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Solution:
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