1. Vision Lab System VISION SYSTEM Chapter 9. Design via Root Locus Youngjoon Han young@ssu.ac.kr

2. Vision Lab System VISION SYSTEM Introduction  Improving Transient Response –Flexibility in the design of a desired transient response can be increased if we can design for transient responses that are not on the root locus –Inserting a differentiator in the forward path in parallel with the gain.

3. Vision Lab System VISION SYSTEM Introduction  Improving Steady-state Error –When the system gain was adjusted to meet the transient response specification, steady-state error performance can be deteriorated  Using dynamic compensators –Steady-state error can be improved by adding an open-loop pole( a pure integrator) at the origin in the forward path  Increasing the system the system type  Driving the associated steady-state error to zero

4. Vision Lab System VISION SYSTEM Introduction  Method to change a system’s root locus –Replacing the existing system with a system whose root locus intersects the design point, B. Expensive and counterproductive –Compensated system having a root locus that goes through the desired pole location for some value of gain. Additional poles and zeros can be added at the low-power end of the system before the plant.

5. Vision Lab System VISION SYSTEM Introduction  Two configurations of compensating Cascade Feedback

6. Vision Lab System VISION SYSTEM Introduction  Compensators –Ideal (active) compensators Compensators that use pure integrator for improving steady- state error or pure differentiator for improving transient response Requiring the use of active amplifiers and possible additional power sources. Steady-state error is reduced to zero. –Passive compensator Requiring the use of passive elements such as resistors and capacitors Less expensive and not requiring additional power source Not driving steady-state error to zero

7. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Method to improve the steady-state error of a feedback control system using cascade compensating –Ideal integral compensation Using a pure integrator to place an open-loop, forward-path pole at the origin Proportional-plus-integral (PI) controller  Increasing the system type and reducing the error to zero –Do not use pure integrator A less expensive passive network that does not requiring additional power sources Lag compensator  Yielding a measurable reduction in steady-state error

8. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Ideal integral Compensation (PI)

9. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Ideal integral Compensation (PI) Approximately on the root locus with compensator pole and zero added

10. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.1 –Given the system operating with a damping ratio of 0.174, show that the addition of the ideal integral compensator reduces the steady-state error to zero for a step input without appreciably affecting transient response. Closed-loop system for Example 9.1: a. before compensation; b. after ideal integral compensation

11. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.1 Root locus for uncompensated system Root locus for compensated system

12. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.1 –Ideal integral compensated system response and the uncompensated system response

13. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  A method of implementing an ideal integral compensator

14. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Lag Compensation Static error constant Improvement in the steady-state error

15. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Lag Compensation –Improving the steady-state error by a factor 10 if the system is operating with a damping ratio of 0.174 Root locus: a. before lag compensation; b. after lag compensation

16. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Lag Compensation –Uncompensated system error is 0.108 with K p = 8.23 –For tenfold improvement in a steady-state error –Required ratio of the compensator zero to the compensator pole

17. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.2

18. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.2

19. Vision Lab System VISION SYSTEM Improving Steady-state Error via Cascade Compensation  Example 9.2

20. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Two method to improve the transient response of a feedback control system –Ideal derivative compensation Addition of a zero to the forward-path transfer function Requiring an active network for its realization Differentiating high-frequency noise yields a large unwanted signal Proportional-plus-derivative (PD) controller –Not using pure differentiation Approximating differentiation a passive network by adding a zero and a more distant pole to the forward-path transfer function Lead compensator

21. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Ideal Derivative Compensation (PD) –Sum of a differentiator and a pure gain –A compensator transfer function –Evaluating the sum of angles from the open-loop poles and zeros to a design point that is the closed- loop pole  Angular contribution of the compensator zero is the difference between 180 o and the calculated angle

22. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Ideal Derivative Compensation (PD) –Sum of a differentiator and a pure gain

23. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Ideal Derivative Compensation (PD)

24. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Ideal Derivative Compensation (PD)

25. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.3 –Design an ideal derivative compensator to yield a 16% overshoot, with a threefold reduction in settling time.

26. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.3 Uncompensated system Compensated dominant pole 

27. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.3 Evaluating the location of the compensation zero -275.6 + 95.6 = -180

28. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.3

29. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.3 Compensated system Uncompensated and compensated system step responses

30. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Implement of the ideal derivative, or PD controller

31. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Lead Compensation –An active ideal derivative compensator can be approximated with a passive lead compensator –The angular contribution of the compensator pole subtracts from the angular contribution of the zero –No additional power supplies are required. –Noise due to differentiation is reduced. –Additional pole does not reduce the number of branches of the root locus that cross the imaginary axis into right half-plane

32. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Lead Compensation

33. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.4 –Design three lead compensators for the system that will reduce the settling time by a factor of 2 while maintaining 30% overshoot. Uncompensated and compensated dominant poles

34. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.4

35. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.4 Compensated system root locus Calculating the location of the compensator pole -172.69 – 7.31 = - 180

36. Vision Lab System VISION SYSTEM Improving Transient Response Via Cascade Compensation  Example 9.4 Uncompensated system and lead compensation responses

37. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Improving Transient Response and Steady-state error  the slight decrease in the speed of the response when the steady- state error is improved  Improving Steady-state error and Transient Response  Improvement in transient response in some cases yields deterioration in the improvement in steady-state response  Proportional-plus-integral-plus-derivative (PID) controller: designing an active PD controller followed by an active PI controller  Lag-lead compensator: designing passive lead compensator followed by a passive lag compensator.

38. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  PID Controller Design

39. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Designing step (1)Evaluate the performance of the uncompensated system to determine how much improvement in transient response is required. (2)Design the PD controller (or lead compensator) to meet the transient response specification. (3)Simulate the system to be sure all requirements have been met. (4)Resign if the simulation show that requirements have not been met. (5)Design the PI controller ( or lag compensator) to yield the required steady-state error. (6)Determine the gains, K1, K2, and K3 of the PID controller ( lag-lead compensator). (7)Simulate the system to be sure all requirements have been met. (8)Resign if simulation shows that requirement have not been met

40. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design –Design a PID controller so that the system can operate with a peak time that is two-thirds that of the uncompensated system at 20% overshoot and with zero steady-state error for a step input

41. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Root locus for uncompensated system Calculating the PD compensator zero Step 1 Step 2  G PD (s) = (S+55.92)

42. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Predicted characteristics of uncompensated, PD-, and PID- compensated systems

43. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Calculating the PD compensator zero Step 2  G PD (s) = (S+55.92) Root locus for PD-compensated system

44. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Step 3 and 4 Step responses for uncompensated, PD- compensated, and PID-compensated Systems

45. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Step 5

46. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.5 PID controller design Step 6  K 1 = 259.5, K 2 =128.6 and K 3 = 4.6 Step 7 and 8

47. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design –Design a lag-lead compensator for the system so that the system will operate with 20% overshoot and a twofold reduction in settling time. Further, the compensated system will exhibit a tenfold improvement in steady-state error for a ramp input.

48. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design The location of the compensator zero coincident with the open-loop at -6. Step 1 and 2 Pc= 29.1

49. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 1 and 2

50. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 3 and 4

51. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 3 and 4 Improvement in step response for lag-lead- compensated System

52. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 5 K v = 3.201 K v = 6.794  Improving the steady-state error by a factor of 2.122  Being designed to improve the steady-state error by a factor of 4.713(10/2.122)

53. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 6 Root locus for lag-lead- compensated system

54. Vision Lab System VISION SYSTEM Improving Steady-State Error and Transient Response  Example 9.6 Lag-lead compensator design Step 7 Improvement in ramp response error for the system Improvement in step response for lag-lead- compensated System

55. Vision Lab System VISION SYSTEM Cascade Compensation

56. Vision Lab System VISION SYSTEM Cascade Compensation

57. Vision Lab System VISION SYSTEM Feedback Compensation  Transfer function designed to be placed in a feedback path can also reshape the root locus.  A generic configuration showing a compensator, H c (s), placed in the minor loop of a feedback control system

58. Vision Lab System VISION SYSTEM Feedback Compensation  More complicated than for cascade compensation  Yielding faster responses  Designing faster responses into portion of a control loop in order to provide isolation  Being used in case where noise problems preclude the use of cascade compensation  Not requiring additional amplification

59. Vision Lab System VISION SYSTEM Feedback Compensation  Finding the gain, such as K, K 1, and K f after establishing a dynamic form for H c (s)  Two approach –Similar to cascade compensation –Designing a specified performance for the minor loop followed by a design of the major loop

60. Vision Lab System VISION SYSTEM Feedback Compensation  Approach 1 –Open loo gain( G(s)H(s) ) –Without feedback( K f H c (s)), loop gain( G(s)H(s) )  Effect of adding feedback is to replace the poles and zeros of G 2 (s) with the poles and zeros of [ K f H c (s) +KG 2 (s)]  Similar to cascade compensation

61. Vision Lab System VISION SYSTEM Feedback Compensation  Example 9.49 –Design rate feedback compensation to reduce the settling time by a factor of 4 while continuing to operate the system with 20% overshoot

62. Vision Lab System VISION SYSTEM Feedback Compensation  Example 9.49 Root locus for uncompensated systemStep Response for uncompensated system

63. Vision Lab System VISION SYSTEM Feedback Compensation  Example 9.49

64. Vision Lab System VISION SYSTEM Feedback Compensation  Example 9.49 –Summing the angle from the uncompensated system’s pole is -277.33 –A compensator zero contribution of + 97.33 to yield 180 – 14.12 /(7.236 – z c ) = tan(180 o -97.33 o )  z c =5.42

65. Vision Lab System VISION SYSTEM Feedback Compensation  Example 9.49 Root locus for the compensated system Step response for compensated system K 1 K f = 256 K f = 1/z c = 0.185  K 1 = 1388

66. Vision Lab System VISION SYSTEM Feedback Compensation  Approach 2 –Using feedback compensation to design a minor loop’s transient response separately from the closed- loop system response. –Poles of forward-path transfer function can be adjusted with the minor-loop gain  This poles then become the open-loop poles for the entire control system

67. Vision Lab System VISION SYSTEM Feedback Compensation  Minor-loop feedback compensation –Design minor-loop feedback compensation to yield a damping ratio of 0.8 for the minor loop and a damping ratio of 0.6 for the closed-loop system.

68. Vision Lab System VISION SYSTEM Feedback Compensation  Minor-loop feedback compensation

69. Vision Lab System VISION SYSTEM Feedback Compensation  Minor-loop feedback compensation

70. Vision Lab System VISION SYSTEM Feedback Compensation  Minor-loop feedback compensation

71. Vision Lab System VISION SYSTEM Physical realization of compensation  Active-circuit Realization

72. Vision Lab System VISION SYSTEM Physical realization of compensation  Active-circuit Realization

73. Vision Lab System VISION SYSTEM Physical realization of compensation  Example 9.9 –Implementing a PID controller  

74. Vision Lab System VISION SYSTEM Physical realization of compensation  passive-circuit Realization

75. Vision Lab System VISION SYSTEM Physical realization of compensation  Lag-lead Compensator Implemented with operational amplifiers