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Chapter 9 Design of Control Systems

Chapter 9 Design of Control Systems. Automatic Control Systems, 9 th Edition F. Golnaraghi & B. C. Kuo. 1, p. 487. 9-1 Introduction. Control system design involves the following 3 steps: Determine what the system should do and how to do it ( design specifications )

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Chapter 9 Design of Control Systems

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  1. Chapter 9Design of Control Systems Automatic Control Systems, 9th Edition F. Golnaraghi & B. C. Kuo

  2. 1, p. 487 9-1 Introduction • Control system design involves the following 3 steps: • Determine what the system should do and how to do it (design specifications) • Determine the controller or compensator configuration, relative to how it is connected to the controlled process • Determine the parameter values of the controller to achieve the design goals

  3. 1, p. 487 Design Specifications • Relative stability • Steady-state accuracy (error) • Transient response characteristics • Maximum overshoot, rise time, settling time,delay time • Frequency response characteristics • Gain margin, phase margin, resonant peak

  4. 1, p. 489 Controller Configurations

  5. 1, p. 490 Controller Configurations (cont.)

  6. 1, p. 490 Controller Configurations (cont.)

  7. 1, p. 491 Fundamental Principles of Design (1/2) • Complex-conjugate poles of the closed-loop transfer function lead to a step response that is underdamped. If all system poles arereal, the step response is overdamped.However, zeros of the closed-loop transfer function may cause overshoot even if the system is overdamped. • The response of a system is dominated by those poles closest the origin in the s-plane.Transients due to those poles farther to left decay faster. • The farther to the left in the s-plane the system’s dominant poles are, the faster the system will respond and the greater its bandwidth will be.

  8. 1, p. 492 Fundamental Principles of Design (2/2) • Pole-zero cancellation:When a pole and zero of a system transfer function nearly cancel each other, the portion of the system response associated with the pole will have a small magnitude. • Time-domain and frequency-domain specifications are loosely associated with each other.Rise time and bandwidth are inversely proportional.Larger phase margin, larger gain margin, and lower resonant peak Mr will improve damping.

  9. 2, p. 492 9-2 Design with the PD Controller PD control adds a simple zero at s = KP/KD to the forward-path transfer function Gp(s)

  10. 2, p. 493 Realizations of PD Controller

  11. 2, p. 495 Time-Domain Interpretation Reduce the overshoot: • 0 < t < t1: the positive correcting torque is too large  decrease • t1 < t < t2: the retarding torque is inadequate increase • t2 < t < t3: the negative correcting torque should be reduced. • t3 < t < t4: the retarding torque should be increased. + + 

  12. 2, p. 496 Effects Provided by PD Controller • 0 < t < t1: de(t)/dt is negativethis will reducethe origin torque developed due to e(t) alone • t1 < t < t2: both e(t) and de(t)/dt are negativethe negative retarding torque developed will be grater than that with only proportional control. • t2 < t < t3: e(t) and de(t)/dt have opposite signsthe negative retarding torque that originally contributes to the undershoot is reduced also. • Derivative or PD control will have an effect on a steady-state error only if the error varies with time. + +   + 

  13. 2, p. 496 Frequency-Domain Interpretation a high-pass filter:accentuate high-frequency noise. • The phase-lead property may be utilized to improve the phase margin of a control system. • The PD controller push the gain-crossover freq. to a higher value.

  14. 2, p. 497 Summary of Effects of PD Control • Improving damping and reducing maximum overshoot • Reducing rise time and settling time • Increasing BW • Improving GM, PM, and Mr • Possibly accentuating noise at higher frequencies • Possibly requiring a relatively large capacitor in circuit implementation

  15. 2, p. 498 Example 9-2-1 Forward-path transfer function: Performance specifications: K = 181.17 (without PD controller) damping ratio: 0.2 maximum shoot: 52.7%

  16. 2, p. 498 Example 9-2-1: Time-Domain Design • Forward-path transfer function: • Closed-loop transfer function: • Ramp-error constant:steady-state error due to a unit-ramp input:

  17. 2, p. 500 Example 9-2-1 (cont.) • Characteristic eq.:

  18. 2, p. 502 Example 9-2-1 (cont.) • Another analytic way:From Eq. (9-14) Applying the stability requirement to Eq. (9-14) 

  19. 2, p. 503 Example 9-2-1 (cont.)

  20. 2, p. 504 Example 9-2-1: Freq.-Domain Design

  21. 2, p. 505 Example 9-2-1 (cont.) Mr = 1.025 Mr = 2.522

  22. 2, p. 505 Example 9-2-1 (cont.) • Performance criteria: • Steady-state error due to a unit-ramp input  0.00443 • Phase margin PM  80° • Resonant peak Mr  1.05 • Bandwidth BW  2000 rad/sec

  23. 2, p. 506 Example 9-2-2 Forward-path transfer function: The same set of time-domain specification. • K = 181.17  maximum overshoot = 78.88% • Forward-path transfer function with PD controller (K = 181.17) • Characteristic equation:

  24. 2, p. 507 Example 9-2-2: Time-Domain Design

  25. 2, p. 508 Example 9-2-2 (cont.) large large small large small

  26. 2, p. 508 Example 9-2-2 (cont.) Improve the damping of the system Does not meet the maximum-overshoot requirement(maximum overshoot  5%)

  27. 2, p. 509 Example 9-2-2: Freq.-Domain Design KP= 1

  28. 2, p. 509 Example 9-2-2 (cont.) KP= 1

  29. 2, p. 510 Example 9-2-2 (cont.) Optimal value

  30. 3, p. 511 9-3 Design with the PI Controller Effects of PI controller: • adding a zero at s = KI/KP to the forward-path transfer function Gp(s) • adding a pole at s = 0 to the forward-path transfer function system type is increased by 1 improve the steady-state error

  31. 3, p. 512 Realizations of PD Controller

  32. 3, p. 514 Frequency-Domain Interpretation a low-pass filter • Gc(j) = 20log10KP dB at  =   attenuation if KP<1 • The phase of Gc(j) is always negative.

  33. 3, p. 514 Frequency-Domain Design Procedure • Made the Bode-plot of the forward-path transfer function of the uncompensated system. • Determine the phase margin and the gain margin.Find the new gain-crossover frequency for a specified phase margin requirement. • Bring the magnitude curve of the uncompensated transfer function down to 0 dB at .

  34. 3, p. 515 Design Procedure (cont.) and Effects • Investigate the Bode plot of the compensated system to see if the performance specifications are all met. • Give the desired transfer function of PI controller Eq. (9-30) Effects of PI controller: • Improving damping and reducing maximum overshoot • Increasing rise time • Decreasing BW • Improving GM, PM, and Mr • Filtering out high-frequency noise

  35. 3, p. 516 Example 9-3-1 Forward-path transfer function: Time-domain performance requirements: • Forward-path transfer function with PI controller: • Parabolic-error constant:

  36. 3, p. 517 Example 9-3-1: Time-Domain Design • Steady-state error:K = 181.17  KI(min) = 0.002215 • Characteristic equation (K = 181.17):Routh’s test  stable: 0 < KI/KP < 361.2 • Place the zero at  KI/KP relatively close to the origin • The most significant pole of GP(s), besides the pole at s = 0, is at 361.2

  37. 3, p. 518 Example 9-3-1 (cont.) •   0.707, KP = 0.08 • KI / KP= 10 KI = 0.8 Three characteristic Eq. roots:s = 10.605, 175.3  j175.4 • KI / KP= 5 KI = 0.4 Three characteristic Eq. roots:s = 5.145, 178.03  j178.03

  38. 3, p. 519 Example 9-3-1 (cont.) meet the requirements increase the rise time reduce the overshoot

  39. 3, p. 520 Example 9-3-1 (cont.)

  40. 3, p. 521 Example 9-3-1: Freq.-Domain Design KP= 1

  41. 3, p. 521 Example 9-3-1 (cont.) KP= 1 868 170

  42. 3, p. 522 Example 9-3-1 (cont.) • KP = 1, KI = 0  PM = 22.68° • Required PM: at least 65° • PM = 65°  new gain-crossover frequency 

  43. 3, p. 523 Example 9-3-1 (cont.) meet the requirements vary little

  44. 3, p. 523 Example 9-3-2 Forward-path transfer function: Time-domain specifications: • Forward-path transfer function with PI controller: • Characteristic equation (K = 181.17) :

  45. 3, p. 524 Example 9-3-2: Time-Domain Design • K = 181.17  KI(min) = 0.002215 • Routh’s tabulation:stability requirements:

  46. 3, p. 524 Example 9-3-2 (cont.)

  47. 3, p. 526 Example 9-3-2 (cont.) The best rise and settling times

  48. 3, p. 525 Example 9-3-2: Freq.-Domain Design K = 181.17 KP = 1 KI= 0

  49. 3, p. 525 Example 9-3-2 (cont.) K = 181.17 KP = 1 KI= 0 163

  50. 3, p. 526 Example 9-3-2 (cont.) • Performance data of uncompensated system:(K = 181.17, KP = 1, KI= 0)GM = 3.578 dB, PM = 7.788°Mr = 6.572, BW = 1378 rad/sec • Required PM: at least 65° • PM = 65°  new gain-crossover frequency  PM with these design parameters is only 59.52  65

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