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Manufacturing Controls

Manufacturing Controls. FALL 2001 Lecture 15 . 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

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Manufacturing Controls

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  1. Manufacturing Controls FALL 2001 Lecture 15  (C) 2001, Ernest L. Hall, University of Cincinnati

  2. 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 Examples Ch. 5 • 13. Nov. 1 Boolean Logic ,Programmable Logic Controllers Ch. 5, Notes • 14. Nov. 6 Stability and Compensators, P, PI and PD Ch. 6 • 15. Nov. 8 PID Controllers - Review 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 • 19. Nov. 22 THANKSGIVING HOLIDAY Ch. 9, Notes • 20. Nov. 27 Optimal Design of a Motion Control System Ch. 9, Notes • 21. Nov. 29 QUIZ 3 (In Class - Closed Book) • 22. Dec. FINAL EXAM (In Class - Closed Book) Comprehensive (C) 2001, Ernest L. Hall, University of Cincinnati

  3. Today’s objective • To continue the introduction to systems theory by reviewing the concepts of control systems for digital control and PID compensation of a feedback control for the motorized arm. • Compensator examples (C) 2001, Ernest L. Hall, University of Cincinnati

  4. Note • Quiz 2 in class next Tuesday (C) 2001, Ernest L. Hall, University of Cincinnati

  5. State space descriptions • State variables are a basic approach to modern control theory. • Mathematically, it is a method for solving an nth order differential equation using an equivalent set of n, simultaneous, first-order differential equations. • Numerically, it is easier to compute solutions to first-order differential equations than for higher-order differential equations. • Practically, it is a way to use digital computers and algorithms based on matrix equations to solve linear or non-linear systems. (C) 2001, Ernest L. Hall, University of Cincinnati

  6. Definitions • A system is described in terms of its state variables, x, which is the smallest set of linearly independent variables that describe the system, • its dynamic state variable, dx/dt, the derivative of the state variable, • its input, u • and its output, y. • A is the system matrix • B is the input matrix • C is the output matrix • D is the feedforward martix (C) 2001, Ernest L. Hall, University of Cincinnati

  7. Mathematical representation • An n-th order differential equation may be rewritten as a set of n simultaneous first order equations. (C) 2001, Ernest L. Hall, University of Cincinnati

  8. Given an n-th order differential equation (C) 2001, Ernest L. Hall, University of Cincinnati

  9. Select a set of state variables called phase-variables as: (C) 2001, Ernest L. Hall, University of Cincinnati

  10. Now differentiate both sides of this equation to obtain the dynamic state equation: (C) 2001, Ernest L. Hall, University of Cincinnati

  11. The last equation may be solved for from the original differential equation to give: (C) 2001, Ernest L. Hall, University of Cincinnati

  12. Now the dynamic state equation may be written: (C) 2001, Ernest L. Hall, University of Cincinnati

  13. These equations may now be put in matrix form: (C) 2001, Ernest L. Hall, University of Cincinnati

  14. Finally the solution to the differential equation may be written as: (C) 2001, Ernest L. Hall, University of Cincinnati

  15. In matrix form • State equation • Output equation (C) 2001, Ernest L. Hall, University of Cincinnati

  16. Example (C) 2001, Ernest L. Hall, University of Cincinnati

  17. Using the phase variables as state variables (C) 2001, Ernest L. Hall, University of Cincinnati

  18. Then the dynamic state equations may be developed using the definition and the system equation (C) 2001, Ernest L. Hall, University of Cincinnati

  19. This is the state variable form for the dynamic system. The output y is simply x1 • These may be written in the linear state equation’s matrix form (C) 2001, Ernest L. Hall, University of Cincinnati

  20. Matrix definitions (C) 2001, Ernest L. Hall, University of Cincinnati

  21. Matlab has built in conversions • ss2tf – state space to transfer function • tf2ss – transfer function to state space (C) 2001, Ernest L. Hall, University of Cincinnati

  22. Example – try this • Num=[24]; • Den=[1,9,26,24]; • [A,B,C,D]=tf2ss(num,den) • A = • -9 -26 -24 • 1 0 0 • 0 1 0 • B = • 1 • 0 • 0 • C = • 0 0 24 • D = • 0 (C) 2001, Ernest L. Hall, University of Cincinnati

  23. Now try ss2tf • [num,den]=ss2tf(A,B,C,D) • num = • 0 0.0000 0.0000 24.0000 • den = • 1.0000 9.0000 26.0000 24.0000 (C) 2001, Ernest L. Hall, University of Cincinnati

  24. Practice exercises • Given a differential equation • Write the transfer function (C) 2001, Ernest L. Hall, University of Cincinnati

  25. Answer (C) 2001, Ernest L. Hall, University of Cincinnati

  26. Given a transfer function write the differential equation (C) 2001, Ernest L. Hall, University of Cincinnati

  27. Answer (C) 2001, Ernest L. Hall, University of Cincinnati

  28. Given a transfer function compute the step response with Matlab (C) 2001, Ernest L. Hall, University of Cincinnati

  29. Answer • Num=[24]; • Den=[1,9,26,24]; • Sys=tf(num,den); • Step(Sys) (C) 2001, Ernest L. Hall, University of Cincinnati

  30. Cascade two transfer function into a single equivalent block diagram (C) 2001, Ernest L. Hall, University of Cincinnati

  31. Answer • 1 (C) 2001, Ernest L. Hall, University of Cincinnati

  32. Compute the transfer function of a closed loop system (C) 2001, Ernest L. Hall, University of Cincinnati

  33. Answer (C) 2001, Ernest L. Hall, University of Cincinnati

  34. Determine if a system is relatively stable using the Matlab margin function. • G(s) = (s+2)/(s+3) • H(s) = 1 (C) 2001, Ernest L. Hall, University of Cincinnati

  35. Answer • Num=[1,2]; • Den=[1,3]; • Margin(Num,Den) • As shown the phase margin is greater than 40 degrees and the gain margin is greater than 10 dB so the system is relatively stable. (C) 2001, Ernest L. Hall, University of Cincinnati

  36. Determine if a system is absolutely stable by examining the pole zero map (C) 2001, Ernest L. Hall, University of Cincinnati

  37. Answer • Num=[24]; • Den=[1,9,-26,24]; • Pzmap(num,den) • As shown the system is absolutely stable since the poles are all in the left half plane. (C) 2001, Ernest L. Hall, University of Cincinnati

  38. Determine the absolute stability of the following system using pole zero map (C) 2001, Ernest L. Hall, University of Cincinnati

  39. Answer • Num=[24]; • Den=[1,-26,24]; • Pzmap(Num,Den) • As shown it is absolutely unstable since it has two poles in the rhp. (C) 2001, Ernest L. Hall, University of Cincinnati

  40. Any questions? (C) 2001, Ernest L. Hall, University of Cincinnati

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