1. Controller Design (to determine controller settings for P, PI or PID controllers) Based on Transient Response Criteria Chapter 12

2. Desirable Controller Features 1.The closed-loop system must be stable. 2.The effects of disturbances are minimized, i.e., good disturbance rejection. 3.Quick and smooth responses to the set-point changes are guaranteed, i.e., good set-point tracking. 4.Off-set is eliminated. 5.Excessive controller action is avoided. 6.The control system is robust, i.e., it is insensitive to changes in operating conditions and to inaccuracies in process model and/or measurements. Chapter 12

4. Simplified Block Diagram

5. B(s) P(s) D(s)

6. Example

7. Chapter 12

8. Example 12.1

9. Chapter 12

11. Alternatives for Controller Design 1.Direct synthesis (DS) method 2.Internal model control (IMC) method 3.Controller tuning relations 4.Frequency response techniques 5.Computer simulation 6.On-line tuning after the control system is installed. Chapter 12

12. Direct Synthesis

13. Direct Synthesis Steps 1.Specify desired closed-loop response (transfer function) 2.Assume process model 3.Solve for controller transfer function

14. Direct Synthesis to Achieve Perfect Control

15. Direct Synthesis to Achieve Finite Settling Time

16. Example

18. Direct Synthesis for Time- Delayed Systems

19. Taylor Series Approximation

20. Example 1

21. Example 2

22. Pade Approximation

23. Example

24. Use the DS design method to calculate PID controller settings for the process: Chapter 12 Example 12.1

25. Consider three values of the desired closed-loop time constant:. Evaluate the controllers for unit step changes in both the set point and the disturbance, assuming that G d = G. Repeat the evaluation for two cases: a.The process model is perfect ( = G). b.The model gain is = 0.9, instead of the actual value, K = 2. Thus, The controller settings for this example are: 3.75 1.88 0.682 8.33 4.17 1.51 15 3.33 Chapter 12

26. Figure 12.3 Simulation results for Example 12.1 (a): correct model gain. Chapter 12

27. Simulation results for Example 21.1(b): incorrect model gain.

28. Chapter 12

29. PID vs. IMC

30. PID Controller Design Procedure Based on IMC Method –Step 1: factor process model

31. PID Controller Design Procedure Based on IMC Method –Step 2: derive IMC transfer function

32. PID Controller Design Procedure Based on IMC Method –Step 3: derive PID transfer function

33. Chapter 12

34. Example

36. Controller Synthesis Criteria in Time Domain Time-domain techniques can be classified into two groups: (a) Criteria based on a few points in the response (b) Criteria based on the entire response, or integral criteria

37. Approach (a) Based on settling time, % overshoot, rise time, decay ratio (Fig. 5.10 can be viewed as closed- loop response). Several methods based on 1/4 decay ratio have been proposed, e.g., Cohen-Coon and Ziegler- Nichols.

38. Chapter 12

40. Approach (b) - Criteria Integral of absolute value of error (IAE) Integral of square error (ISE) Time-weighted IAE (ITAE)

41. Approach (b) - Remarks Pick controller parameters to minimize integral. 1.IAE allows larger overall deviation than ISE (with smaller overshoots). 2.ISE needs longer settling time 3.ITAE weights errors occurring later more heavily Approximate optimum tuning parameters are correlated with K, ,  (Table 12.3).

42. Chapter 12

46. Example 1

47. ITAE0.1691.85 IAE0.1952.02 ISE0.2452.44

49. Example 2

50. Chapter 12

51. Summary of Tuning Relationships 1. K C is inversely proportional to K P K V K M. 2. K C decreases as  /  increases. 3.  I and  D increase as  /  increases (typically  D = 0.25  I ). 4. Reduce K c, when adding more integral action; increase K c, when adding derivative action 5. To reduce oscillation, decrease K C and increase  I. Chapter 12

52. Disadvantages of Tuning Correlations 1. Interactions are ignored (decreased stability limits). 2. Derivative action equipment specific. 3. First order + time delay model can be inaccurate. 4. K p,  can vary. 5. Resolution, measurement errors decrease stability margins. 6. ¼ decay ratio not conservative standard (too oscillatory). Chapter 12

53. Example 12.4 Consider a lag-dominant model with Design four PI controllers: a)IMC b)IMC based on the integrator approximation in Eq. 12- 33 c)IMC with Skogestad’s modification (Eq. 12-34) d)Direct Synthesis method for disturbance rejection (Chen and Seborg, 2002): The controller settings are K c = 0.551 and Chapter 12

54. Evaluate the four controllers by comparing their performance for unit step changes in both set point and disturbance. Assume that the model is perfect and that G d (s) = G(s). Solution The PI controller settings are: ControllerKcKc (a)IMC0.5100 (b) Integrator approximation0.556 5 (c) Skogestad0.5 8 (d) DS-d0.551 4.91 Chapter 12

55. Figure 12.8. Comparison of set-point responses (top) and disturbance responses (bottom) for Example 12.4. The responses for the Chen and Seborg and integrator approximation methods are essentially identical. Chapter 12

57. On-Line Controller Tuning 1.Controller tuning inevitably involves a tradeoff between performance and robustness. 2.Controller settings do not have to be precisely determined. In general, a small change in a controller setting from its best value (for example, ±10%) has little effect on closed-loop responses. 3.For most plants, it is not feasible to manually tune each controller. Tuning is usually done by a control specialist (engineer or technician) or by a plant operator. Because each person is typically responsible for 300 to 1000 control loops, it is not feasible to tune every controller. 4.Diagnostic techniques for monitoring control system performance are available. Chapter 12

58. Controller Tuning and Troubleshooting Control Loops Chapter 12

59. Ziegler-Nichols Rules: These well-known tuning rules were published by Z-N in 1942: controllerKcKc II DD P PI PID 0.5 K CU 0.45 K CU 0.6 K CU - P U /1.2 P U /2 - P U /8 Z-N controller settings are widely considered to be an "industry standard". Z-N settings were developed to provide 1/4 decay ratio -- too oscillatory?

60. Chapter 12 Modified Z-N settings for PID control controllerKcKc II DD original Some overshoot No overshoot 0.6 K CU 0.33 K CU 0.2 K CU P U /2 P U /3 P U /8 P U /3 P U /2

61. Chapter 12

63. Figure 12.15 Typical process reaction curves: (a) non-self- regulating process, (b) self-regulating process. Chapter 12

64. Figure 12.16 Process reaction curve for Example 12.8. Chapter 12

65. Figure 12.17 Block diagram for Example 12.8. Chapter 12