MECH 420: Fundamentals of Control Systems

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MECH 420 Fundamentals of Control Systems

• Ali Israr (post doc)Mechatronics and Haptic
  Interface Lab
• Rice University
• January 28, 2009

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Content Outline

• Feedback Control Systems
• Time and Laplace Representation
• Block Diagram Algebra
• Closed-loop Transfer Function
• Effects of Disturbances

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Feedback Systems

• Open loop
• Closed loop Example Cruise control, auto pilot
  etc.
• Achieve required response, even with uncertainty
  in model
• Robust to noise disturbance
• Stability

• Dynamics are known and well modeled
• No uncertainty or external disturbance

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Temporal vs. Laplace Domain

• Temporal domain
• Laplace domain
• Temporal Laplace
• Convolution Multiplication

Stable??
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Transfer Function

• Only for LTI systems
• Inherent property of the system does not depend
  on the external inputs
• Denominator represents stability
• Commutative
• Associative

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Transfer Function
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Block Diagram Algebra
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Example 1 Y(s)/R(s)? (1 of 6 )
Transformation 4 (outside block coming in the
loop)
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Example Y(s)/R(s)? (cont. 2 of 6)
Transformation 6
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Example Y(s)/R(s)? (cont. 3 of 6)
Transformation 1 (Cascade blocks)
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Example Y(s)/R(s)? (cont. 4 of 6)
Transformation 6
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Example Y(s)/R(s)? (cont. 5 of 6)
Transformation 1 (Cascade blocks)
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Example Y(s)/R(s)? (cont. 6 of 6)
c
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Example 1 (Summary)
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Class Exercise
Is the overall transfer function stable?
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Class Exercise (Worksheet)
Stable???
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Class Exercise 2
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Class Exercise 2 (cont.)
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Class Example 2
Find
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Class Example 2 (part 1)
Find
Alternate method
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Class Example 2 (part 2)
Find
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Class Example 2 (part 2)
Find
Alternate method
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Class Example 2 (part 3)
Find
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Class Example 2 (part 3)
Find
Alternate method
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Class Example 2 (part 4)
Find
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Class Example 2 (part 4)
Find
Alternate method
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Exercise

• What are the poles and zeros of the process?
• Is the process stable?
• Find the overall transfer function Y/R.
• Is the overall transfer function Y/R stable?
  Why?
• Find the transfer function Y/D0.
• In the presence of disturbance, is the system
  response stable? Why?

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Exercise

• What are the poles and zeros of the process?
• z-5 p7 and -2
• Is the process stable?
• No.

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Exercise

• Find the overall transfer function Y/R.
• Is the overall transfer function Y/R stable? Why?
• Yes. Poles at -19.94, -7.03, -2.03.
• All stable poles and zero-pole cancellation
  of unstable pole

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Exercise

• Find the transfer function Y/D0.
• In the presence of disturbance, is the system
  response stable? Why?
• NO! Unstable pole sign change in the
  denominator.

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Thank you!!