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Chapter 9 Design Via Root Locus

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Chapter 9 Design Via Root Locus

  1. 1. CHAPTER 9 DESIGN VIA ROOT LOCUS [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  2. 2. INTRODUCTION <ul><li>Objectives: </li></ul><ul><ul><li>How to use the root locus to design cascade compensators to improve the steady state error </li></ul></ul><ul><ul><li>How to use the root locus to design cascade compensators to improve the transient response </li></ul></ul><ul><ul><li>How to use the root locus to design cascade compensators to improve both the steady state error and the transient response </li></ul></ul><ul><ul><li>How to use the root locus to design feedback compensators to improve the transient response </li></ul></ul><ul><ul><li>How to realize the designed compensators physically </li></ul></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  3. 3. INTRODUCTION [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  4. 4. IMPROVING TRANSIENT RESPONSE <ul><li>Rather than replacing the existing system with a system whose root locus intersects the desired design of point B, we can augment or compensate the system with additional poles and zeros – so that the compensated system has a root locus that goes through the desired pole location for some value of gain </li></ul><ul><li>Two methods – passive or active network </li></ul><ul><li>Disadvantages – system order can increase, subsequently effect the desired response </li></ul><ul><li>One method of compensating for transient response is to insert differentiator in the forward path in parallel with the gain </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  5. 5. IMPROVING STEADY STATE ERROR <ul><li>Compensators are not only used to improve the transient response of a system, they are also used independently to improve steady state error characteristics </li></ul><ul><li>In Chapter 7 – steady state error can be improved by adding an open loop pole at the origin in forward path, thus increasing the system type and driving the associated steady state error to zero </li></ul><ul><li>This additional pole at the origin requires an integrator for its realization </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  6. 6. Compensators [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  7. 7. Configuration of Compensations [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  8. 8. System Improvement Technique <ul><li>Feeding method </li></ul><ul><ul><li>Proportional – feed the error forward to the plant </li></ul></ul><ul><ul><li>Integral – feed the integral of the error to the plant </li></ul></ul><ul><ul><li>Derivative – feed the derivative of the error to the plant </li></ul></ul><ul><li>Implemented using active networks (PI/PD) </li></ul><ul><ul><li>Using Amplifiers </li></ul></ul><ul><ul><li>Expensive </li></ul></ul><ul><li>Implemented using passive networks (Lag/lead) </li></ul><ul><ul><li>Less expensive </li></ul></ul><ul><ul><li>no additional power required </li></ul></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  9. 9. IMPROVING STEADY-STATE ERROR VIA CASCADE COMPENSATION [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  10. 10. IMPROVING TRANSIENT RESPONSE VIA CASCADE COMPENSATION [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  11. 11. IMPROVING STEADY-STATE ERROR AND TRANSIENT RESPONSE [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  12. 12. PI, PD, PID CONTROLLER [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  13. 13. The Characteristics of P, I & D Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  14. 14. Mathematical Representation of Proportional Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  15. 15. Mathematical Representation of Integral Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  16. 16. Mathematical Representation of Derivative Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  17. 17. Mathematical Representation of PI Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  18. 18. More on IDEAL INTEGRAL COMPENSATION (PI Controller) [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  19. 19. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka - θ 1 - θ 2 - θ 3 - θ pc + θ zc ≡ (2k+1)180 0
  20. 20. PI controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  21. 21. EXAMPLE: Closed-loop system a. before compensation; b. after ideal integral compensation [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  22. 22. Root locus for uncompensated system [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  23. 23. Root locus for compensated system [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  24. 24. Ideal integral compensated system response and the uncompensated system response of previous example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  25. 25. LAG COMPENSATION <ul><li>If use passive networks, the pole and zero are moved to the left, close to the origin </li></ul><ul><li>This placement usually will not increase the system type, but will yield an improvement in the static error constant over an uncompensated system </li></ul><ul><li>Although the ideal compensator drives the steady state error to zero, a lag compensator with a pole that is not at the origin will improve the static error constant by a factor equal to Z c /P c </li></ul><ul><li>There will also minimal effect upon the transient response if the pole-zero pair of the compensator close to the origin </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  26. 26. a. Type 1 uncompensated system; b. Type 1 compensated system; c. compensator pole-zero plot [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  27. 27. Root locus: a. before lag compensation; b. after lag compensation [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  28. 28. EXAMPLE [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  29. 29. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  30. 30. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  31. 31. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  32. 32. Mathematical Representation of PD Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  33. 33. More on IDEAL DERIVATIVE COMPENSATION (PD Controller) <ul><li>To speed up the original system, we can add a single zero to the forward path </li></ul><ul><li>This zero can be represented by a compensator whose transfer function is </li></ul><ul><ul><ul><li>G c (S) = s + Z c </li></ul></ul></ul><ul><li>This function, the sum of a differentiator and a pure gain is called ideal derivative – PD controller </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  34. 34. PD controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  35. 35. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka Using ideal derivative compensation: a. uncompensated; b. compensator zero at –2; c. compensator zero at –3; d. compensator zero at – 4
  36. 36. <ul><li>Zero is moved to a different position and for each compensated case, the dominant, second order poles are farther out along the 0.4 damping ratio line than the uncompensated system </li></ul><ul><li>However, each of the compensated case has dominant poles with the same damping ratio as the uncompensated case  thus the percent overshoot is predicted to be similar for each case! </li></ul><ul><li>The compensated dominant, closed loop poles has more negative real parts than the uncompensated dominant, closed loop poles  thus shorter settling time, T s </li></ul><ul><li>On top of that smaller peak time, T p </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  37. 37. Predicted characteristics for the previous shown systems [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  38. 38. Uncompensated system and ideal derivative compensation solutions from previous table [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  39. 39. Example: Feedback control system for Example 9.3 [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  40. 40. Root locus for uncompensated system shown in Previous Example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  41. 41. Uncompensated and compensated system characteristics for Previous example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  42. 42. Compensated dominant pole superimposed over the uncompensated root locus for previous example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  43. 43. Evaluating the location of the compensating zero [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  44. 44. Root locus for the compensated system [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  45. 45. Uncompensated and compensated system step responses of previous example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  46. 46. LEAD COMPENSATION <ul><li>Similar to the active ideal integral that can be approximated with passive lag network, an active ideal integral can be approximated with a passive lead compensator </li></ul><ul><li>Pole is farther from the imaginary axis then the zero – will result in a positive angular distribution of the compensator and thus approximates an equivalent single zero </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  47. 47. Geometry of lead compensation [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  48. 48. Three of the infinite possible lead compensator solutions [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  49. 49. EXAMPLE [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  50. 50. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  51. 51. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  52. 52. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  53. 53. Other Example [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  54. 54. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  55. 55. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  56. 56. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  57. 57. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  58. 58. Mathematical Representation of PID Controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  59. 59. PID Controller Design <ul><li>It has two zeros and a pole at the origin </li></ul><ul><li>One zero and the pole at the origin can be designed as the ideal integral compensator </li></ul><ul><li>Another zero can be designed as the ideal derivative compensator </li></ul><ul><li>Follow steps in Text book – page 532 </li></ul><ul><ul><ul><li>Evaluate – Design PD – Design PI </li></ul></ul></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  60. 60. PID controller [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  61. 61. LAG-LEAD COMPENSATOR DESIGN <ul><li>First design the lead compensator to improve the transient response </li></ul><ul><li>Next, evaluate the improvement in steady state </li></ul><ul><li>Finally, design the lag compensator to meet the steady state error requirement </li></ul><ul><li>Follow steps in Text Book – Page 537 </li></ul><ul><ul><ul><li>Evaluate – Design Lead – Design Lag </li></ul></ul></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  62. 62. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  63. 63. [email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  64. 64. FEEDBACK COMPENSATION <ul><li>Approach 1 </li></ul><ul><li>Approach 2 </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka
  65. 65. PHYSICAL REALIZATION OF COMPENSATION <ul><li>Active circuit Realization </li></ul><ul><li>Passive circuit Realization </li></ul>[email_address] Fakulti Kejuruteraan Pembuatan Universiti Teknikal Malaysia Melaka

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