Manufacturing Controls

1
Manufacturing Controls

• FALL 2001
• Lecture 11 

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Syllabus

• DATE TOPIC NOTES
• 1. Sep. 20 Mechatronics Design Process Ch. 1
• 2. Sep. 25 System Modeling and Simulation Ch. 2
• 3. Sep. 27 Laplace Transforms and Transfer
  Functions Ch. 2
• 4. Oct. 2 Electrical Examples Ch.2, Notes
• 5. Oct. 4 Mechanical Examples Ch.2, Notes
• 6. Oct. 9 More Examples, Thermal and Fluid
  Examples, QUIZ 1 (Take Home)
• 7. Oct. 11 Sensors and Transducers Ch. 3
• 8. Oct. 16 Digital control, Advanced MATLAB
• 9. Oct. 18 Analog and Digital Sensing Ch. 3,
  Notes
• 10. Oct. 23 Actuating Devices, time and frequency
  response Ch. 4
• 11. Oct. 25 DC Motor Model Ch. 4,
  Notes
• 12. Oct. 30 Boolean Logic Ch. 5
• 13. Nov. 1 Programmable Logic Controllers Ch.
  5, Notes
• 14. Nov. 6 Stability and Compensators, P, PI and
  PD Ch. 6
• 15. Nov. 8 PID Controllers Ch. 7
• 16. Nov. 13 QUIZ 2 (In Class - Open Book)
• 17. Nov. 15 Practical and Optimal Compensator
  Design Ch. 8
• 18. Nov. 20 Frequency Response Methods Ch. 9,
  Notes

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Todays objective

• To continue the introduction to systems theory by
  continuing the concept of frequency response for
  digital control for compensation of a feedback
  control for the motorized arm.
• By the end of this class you will be able to
  describe the advantage of using both time and
  frequency response for describing both the
  actuating device and the control compensation of
  a motorized system.
• Understand first and second order systems.

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Example

• Suppose a DC motor is used to drive a robot arm
  horizontally.

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Frequency Response and Time response

• Permit descriptions with greater clarity
• Time response also important
• Need both

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Sinusoids are eigenvectors of linear systems

• That is, if a sinusoid is put into a linear
  system a sinusoid will be the output. It may be
  changed only in magnitude and phase.

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To get frequency response

• Substitute sjw into transfer function to get
  frequency response

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Time response

• The output of a system is the sum of two
  responses
• Forced response or steady state response or
  particular solution
• Natural response or homogeneous solution

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Second order systems general form
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Stable forms of second order system

• Look at signs of coefficients with agt0 and bgt 0
• Which of these are absolutely stable, i.e. have
  poles in the left half plane?

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Routh-Hurwitz Criterion

• A systematic method for determining if the
  characteristic equation has poles in the rhp, lhp
  or on the jw axis is the Routh Hurwitz criteria.
• It requires making a table
• Examining the table for sign changes in the first
  colums

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Example for a4s4a3s3a2s2a1sa0
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Second column
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Third column
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Example unity feedback system
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First step- form closed loop response
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Example for s310s231s1030
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Any row can be multiplied by a positive
constant without changing the results
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Computation
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Basic rule

• The number of roots of the polynomial that are in
  the right half plane is equal to the number of
  sign changes in the first column.
• In this example, there are two sign changes
  indicating two poles in the right half plane and
  an unstable system.

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Check with Matlab

• num 0,0,0,1
• den 1,10,31,1030
• systf(num,den)
• pzmap(sys)

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Back to second order system
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Stable forms of second order system

• Look at signs of coefficients with agt0 and bgt 0
• Which of these are absolutely stable, i.e. have
  poles in the left half plane?

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Construct Routh Hurwitz table
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Conclusion

• Either a or b or both lt 0 will cause at least one
  sign change in the first column and lead to an
  unstable system
• Only stable second order systems are

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Second order systems

• Response determined by pole locations roots of
  characteristic equation
• Real unequal poles overdamped system
• Complex roots on jw axis Undamped system
• Complex roots not on jw axis Underdamped system
• Real and equal roots Critically damped system

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Consider the step response poles real and
unequal - overdamped from Nice pp. 168
num 0,0,1 den 1,9,9 systf(num,den) pzm
ap(sys) Poles at -7.854 and -1.146
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Now poles complex conjugates - underdampedfrom
Nice pp. 168
num 0,0,1 den 1,2,9 systf(num,den) pzm
ap(sys)
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Now poles complex conjugates on jw axis-
undampedfrom Nice pp. 168
num 0,0,1 den 1,0,9 systf(num,den) pzm
ap(sys)
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Now poles real and equal- critically dampedfrom
Nice pp. 168
num 0,0,1 den 1,6,9 systf(num,den) pzm
ap(sys)
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Summary for second order systems

• Overdamped response sluggish
• Poles two real poles at s1, -s2
• Natural response two exponentials with time
  constants equal to the reciprocal of the pole
  locations

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Underdamped response

• Poles Two complex poles at -sd and jwd
• Natural response Damped sinusoid with an
  exponential envelope whose time constant is equal
  to the reciprocal of the poles real part
• The radian frequency of the sinusoid is equal to
  the imaginary part of the poles

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Undamped response

• Poles Two imaginary poles at and wl
• Natural response Undamped sinusoid with radian
  frequency equal to the imaginary part of the
  poles

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Critically damped response

• Poles Two real, equal poles at sl
• Natural response One term is an exponential
  whose time constant is the reciprocal of the pole
  location. Another term is the product of time, t,
  and an exponential whose time constant is equal
  to the reciprocal of the pole location.
• Note this is the fastest response without
  overshoot.

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General second order system

• Second order systems are so common that a special
  notation has been adopted to describe them.

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Homework due next Thursday, Nov. 1, 2001

• For the following systems,
• Determine the pole-zero plot
• Step response
• And determine if the systems are
• Overdamped
• Underdamped
• Undamped
• Critically damped
• Determine the damping ratio and natural frequency
  when appropriate for each

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Homework systems
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Any questions?