ME375 Dynamic System Modeling and Control

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CHAPTER 4MESB 374 System Modeling and
AnalysisModel Reduction(Electro-mechanical
Systems)
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How do we determine two systems are identical?
Structures and parameters are exactly same
Responses to all inputs are the same

• Step responses are same

Frequency responses are the same
Bode plots are the same
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Definition and Motivations

• Simplification of system model without losing
  main features of system

Motivations Complicated illustration of system
may be distractive Catastrophe, or
Tragedy Sometimes, too detailed modeling may not
be necessary 1.999999 or 2 The cost of
modeling Complicated model may not be suitable
for controller design Cost of implementation of
controller
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One Example
Unit-Step Response of a system with transfer
function
Unit-Step Response of a system with transfer
function
Any detectable Differences?
Poles
Poles
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One Example (continued)
Bode Plots
Bode Plots
Any detectable Differences?
Is there any difference between their responses
if input is
?
When can we use the simpler one?
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Model Reduction of DC Motor

• Original Block Diagram

Transfer Functions
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Block Diagram Reduction
Block diagram has been simplified. Detailed
structures of system have been hidden.
Is this complicated model always necessary?
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Example

• (B) Find the poles of the transfer function.
• (C) Plot the Bode diagram of the transfer function

• (A) Given the following specification of a DC
  motor and assume there is no load, find its
  transfer function from input voltage to motor
  angular speed
• LA 2 mH
• RA 10 W
• KT 0.06 Nm/A
• JA 5 10-6 Kg m2
• B 3 10-6 Nm/(rad/sec)

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Example
S2
S1
Phase (deg) Magnitude (dB)
S2
S1
Q If we are only interested in the system
response up to 400 rad/sec, can we simplify our
model ? How would you simplify the model ?
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If frequency is smaller than 400 rad/sec, the
characteristics of Bode plots is similar to that
of a first-order system. Hence, we can use a
first-order system to approximate it.

• Simplify the transfer function of whole system

• Simplify each block and obtain TF of the
  resulting system

TF
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Model Reduction

• QPhysically, what do we mean by neglecting
  armature inductance ?
• By neglecting the armature inductance, we are
  assuming that it takes no time for the current to
  reach its steady state value when there is a step
  change in input voltage, i.e. a sudden change in
  input voltage will result in a sudden change in
  the armature current, which in turns will result
  in a sudden change in the motor torque output.
  This is equivalent to have direct control of the
  motor current.