1. RELATIVE GAIN MEASURE OF INTERACTION We have seen that interaction is important. It affects whether feedback control is possible, and if possible, its performance. Do we have a quantitative measure of interaction? The answer is yes, we have several! Here, we will learn about the RELATIVE GAIN ARRAY. Our main challenge is to understand the correct interpretations of the RGA.

2. RELATIVE GAIN We are here, and making progress all the time! Defining control objectives Controllability & Observability Interaction & Operating window The Relative Gain Multiloop Tuning Performance and the RDG SVD and Process directionality Robustness Integrity Control for profit Optimization-based design methods Process design - Series and self-regulation - Zeros (good/bad/ugly) - Recycle systems - Staged systems

3. RELATIVE GAIN MEASURE OF INTERACTION Let’s start here to build understanding OUTLINE OF THE PRESENTATION 1.DEFINITION OF THE RGA 2.EVALUATION OF THE RGA 3.INTERPRETATION OF THE RGA 4.EXTENSIONS OF RGA 5.PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA

4. RELATIVE GAIN MEASURE OF INTERACTION ++ + + G 11 (s) G 21 (s) G 12 (s) G 22 (s) G d2 (s) G d1 (s) D(s) CV 1 (s) CV 2 (s)MV 2 (s) MV 1 (s) The relative gain between MV j and CV i is ij. It is defined in the following equation. What have we assumed about the other controllers? Explain in words.

5. RELATIVE GAIN MEASURE OF INTERACTION Now, how do we determine the value? OUTLINE OF THE PRESENTATION 1.DEFINITION OF THE RGA 2.EVALUATION OF THE RGA 3.INTERPRETATION OF THE RGA 4.EXTENSIONS OF RGA 5.PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA

6. RELATIVE GAIN MEASURE OF INTERACTION The relative gain array is the element-by-element product of K with K -1. (  = product of ij elements, not normal matrix multiplication) 1. The RGA can be calculated from open- loop gains (only). Open-loop Closed-loop

7. RELATIVE GAIN MEASURE OF INTERACTION 1. The RGA can be calculated from open- loop values. The relative gain array for a 2x2 system is given in the following equation. What is true for the RGA to have 1’s on diagonal?

8. RELATIVE GAIN MEASURE OF INTERACTION 2. The RGA elements are scale independent. Original unitsModified units What is the effect of changing the units of the CV, expressing CV as % of instrument range, or changing the capacity of the final element on ij ? Can we prove that this is general?

9. RELATIVE GAIN MEASURE OF INTERACTION 3. The rows and columns of the RGA sum to 1.0. For a 2x2 system, how many elements are independent?

10. RELATIVE GAIN MEASURE OF INTERACTION 3. The rows and columns of the RGA sum to 1.0. Class exercise: prove this statement. Hint: a matrix and its inverse commute, i.e., K K -1 = K -1 K = I

11. RELATIVE GAIN MEASURE OF INTERACTION 3. The rows and columns of the RGA sum to 1.0. K K -1 = I = K -1 K From the left hand equation, the elements of I are equal to From the right hand equation, the elements of I are equal to

12. RELATIVE GAIN MEASURE OF INTERACTION 4. In some cases, the RGA is very sensitive to small errors in the gains, K ij. When is this equation very sensitive to errors in the individual gains?

13. RELATIVE GAIN MEASURE OF INTERACTION 4. In some cases, the RGA is very sensitive to small errors in the gains, K ij. We must perform a thorough study to ensure that numerical derivatives are sufficiently accurate! The  x must be sufficiently small (be careful about roundoff). From McAvpy, 1983

14. RELATIVE GAIN MEASURE OF INTERACTION 4. In some cases, the RGA is very sensitive to small errors in the gains, K ij. We must perform a through study to ensure that numerical derivatives are sufficiently accurate! The convergence tolerance must be sufficiently small. Average gains from +/- From McAvpy, 1983

15. RELATIVE GAIN MEASURE OF INTERACTION 5. The relative gain elements are independent of the control design for the “ij” inputs and outputs being considered. AC TC Solvent Reactant F S >> F R

16. RELATIVE GAIN MEASURE OF INTERACTION 6. A permutation in the gain matrix (changing CVs and MVs) results in the same permutation in the RG Array. Process gain RGA ??

17. RELATIVE GAIN MEASURE OF INTERACTION How do we use values to evaluate behavior? OUTLINE OF THE PRESENTATION 1.DEFINITION OF THE RGA 2.EVALUATION OF THE RGA 3.INTERPRETATION OF THE RGA 4.EXTENSIONS OF RGA 5.PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA

18. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij < 0 In this case, the steady-state gains have different signs depending on the status (auto/manual) of the other loops. A A C A0 CACA CSTR with A  B Solvent A Discuss interaction and RGA in this system.

19. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij < 0 In this case, the steady-state gains have different signs depending on the status (auto/manual) of other loops We can achieve stable multiloop feedback by using the sign of the controller gain that stabilizes the multiloop system. Discuss what happens when the other interacting loop is placed in manual!

20. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij < 0 the steady-state gains have different signs For ij < 0, one of three BAD situations occurs 1.Multiloop is unstable with all in automatic. 2.Single-loop ij is unstable when others are in manual. 3.Multiloop is unstable when loop ij is manual and other loops are in automatic

21. FR  XB FV  XD Example of pairing on a negative RGA (-5.09). XB controller has a Kc with opposite sign from single-loop control! The system goes unstable when a constraint is encountered. But, we can achieve stable control with pairing on negative RGA! FR FV XB XD

22. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij < 0 the steady-state gains have different signs For ij < 0, one of three situations occurs 1.The process g ij (s) has a RHP zero 2. The overall plant has a RHP zero 3. The system with g ij (s) removed has a RHP zero See Skogestad and Postlethwaite, 1996

23. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij = 0 In this case, the steady-state gain is zero when all other loops are open, in manual. T L Could this control system work? What would happen if one controller were in manual? Heating tank without boiling

24. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i 0< ij <1 In this case, the multiloop (ML) steady-state gain is larger than the single-loop (SL) gain. What would be the effect on tuning of opening/closing the other loop? Discuss the case of a 2x2 system paired on ij = 0.1

25. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij = 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. ++ + + G 11 (s) G 21 (s) G 12 (s) G 22 (s) G d2 (s) G d1 (s) D(s) CV 1 (s) CV 2 (s)MV 2 (s) MV 1 (s) What is generally true when ij = 1 ? Does ij = 1 indicate no interaction?

26. RELATIVE GAIN MEASURE OF INTERACTION ij = 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. AC TC CSTR with zero heat of reaction Solvent Reactant F S >> F R Determine the relative gain. Discuss interaction in this system.

27. RELATIVE GAIN MEASURE OF INTERACTION ij = 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. 0 0 0 0 0 Lower diagonal gain matrix Diagonal gain matrix Diagonal gain matrix Both give an RGA that is diagonal!

28. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i 1< ij In this case, the steady-state multiloop (ML) gain is smaller than the single-loop (SL) gain. What would be the effect on tuning of opening/closing the other loop? Discuss a the case of a 2x2 system paired on ij = 10.

29. FR  XD FRB  XB FD  XD FRB  XB RELATIVE GAIN MEASURE OF INTERACTION 1. Do level loops affect the composition RGA’s? 2.Does the process operation affect RGA’s? Rel. vol = 1.2, R = 1.2 R min From McAvpy, 1983

30. RELATIVE GAIN MEASURE OF INTERACTION MV j  CV i ij =  In this case, the gain in the ML situation is zero. We conclude that ML control is not possible. How can we improve the situation? Have we seen this result before?

31. RELATIVE GAIN MEASURE OF INTERACTION OUTLINE OF THE PRESENTATION 1.DEFINITION OF THE RGA 2.EVALUATION OF THE RGA 3.INTERPRETATION OF THE RGA 4.EXTENSIONS OF RGA 5.PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA Let’s extend the concept

32. RELATIVE GAIN MEASURE OF INTERACTION The relative gain between MV j and CV i is ij. The basic definition involves steady-state gain information. Some plants are unstable Control performance is influenced by dynamics Many plants have an unequal number of MVs and CVs Control design involves structures other than single-loop Disturbances are not considered!

33. RELATIVE GAIN MEASURE OF INTERACTION We can evaluate the RGA of a system with integrating processes, such as levels. Redefine the output as the derivative of the level; then, calculate as normal. (Note that L is unstable, but dL/dt is stable.) m2m2 m1m1 m L A  = density D = density

34. RELATIVE GAIN MEASURE OF INTERACTION We can evaluate the RGA of a system with integrating processes, such as levels. Redefine the output as the derivative of the level; then, calculate as normal. m2m2 m1m1 m L A  = density D = density

35. FR  XD FRB  XB RELATIVE GAIN MEASURE OF INTERACTION A frequency-dependent RGA can be calculated using the transfer functions in place of the steady-state gains. ++ + + G 11 (s) G 21 (s) G 12 (s) G 22 (s) G d2 (s) G d1 (s) D(s) CV 1 (s) CV 2 (s)MV 2 (s) MV 1 (s) We can evaluate the RGA of dynamics processes

36. Bode plots of the individual transfer functions for a distillation tower

37. Bode plot of the RGA 11 element. What frequency range is most important for feedback control?

38. RELATIVE GAIN MEASURE OF INTERACTION The basic definition involves steady-state gain information. Some plants are unstable Control performance is influenced by dynamics Many plants have an unequal number of MVs and CVs Control design involves structures other than single-loop Disturbances are not considered! Apparently, there is a lot more to learn. We better plan to address these issues in the remainder of the course

39. RELATIVE GAIN MEASURE OF INTERACTION Let’s evaluate some design guidelines based on RGA OUTLINE OF THE PRESENTATION 1.DEFINITION OF THE RGA 2.EVALUATION OF THE RGA 3.INTERPRETATION OF THE RGA 4.EXTENSIONS OF RGA 5.PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA

40. RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #1 Select pairings that do not have any ij <0 Review the interpretation, i.e., the effect on behavior. What would be the effect if the rule were violated? Do you agree with the Proposed Guideline?

41. RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #2 Select pairings that do not have any ij =0 Review the interpretation, i.e., the effect on behavior. What would be the effect if the rule were violated? Do you agree with the Proposed Guideline?

42. RELATIVE GAIN MEASURE OF INTERACTION We conclude that the RGA provides excellent insight into the INTEGRITY of a multiloop control system. INTEGRITY: A multiloop control system has good integrity when after one loop is turned off, the remainder of the control system remains stable. “Turning off” can occur when (1) a loop is placed in manual, (2) a valve saturates, or (3) a lower level cascade controller no lower changes the valve (in manual or reached set point limit). Pairings with negative or zero RGA’s have poor integrity RGA and INTEGRITY

43. RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #3 Select a pairing that has RGA elements as close as possible to ij =1 Review the interpretation, i.e., the effect on behavior. What would be the effect if the rule were violated? Do you agree with the Proposed Guideline?

44. FR  XD FRB  XB FD  XD FRB  XB RGA = 6.09 RGA = 0.39 For set point response, RGA closer to 1.0 is better

45. FR  XD FRB  XB FD  XD FRB  XB RGA = 6.09 RGA = 0.39 For feed composition disturbance response, RGA farther from 1.0 is better

46. RELATIVE GAIN MEASURE OF INTERACTION Using guidelines #1 and #2, the control possibilities for this example process were reduced from 36 to 4.

47. RELATIVE GAIN MEASURE OF INTERACTION Tells us about the integrity of multiloop systems and something about the differences in tuning as well. Uses only gains from feedback process! Does not use following information - Control objectives - Dynamics - Disturbances Lower diagonal gain matrix can have strong interaction but gives RGAs = 1 Can we design controls without this information? The RGA gives useful conclusions from S-S information, but not enough to design process control Powerful results from limited information! “Interaction?”

48. Workshop on Relative Gain Array INTERACTION IN FEEDBACK SYSTEMS

49. Workshop on Relative Gain Array: Problem 1 The RGA has been evaluated, but the regulatory control system (below the loops being analyzed using RGA) has been modified. Instead of adjusting a valve directly, one of the loops being evaluated will adjust a flow controller set point (which adjusts the same valve). How would you evaluate the new RGA?

50. Workshop on Relative Gain Array: Problem 2 You have decided to pair on a loop that has a negative RGA element. Discuss the tuning that is appropriate for this loop. A A C A0 CACA CSTR with A  B Solvent A

51. Workshop on Relative Gain Array: Problem 3 Discuss what information can be obtained from the RGA and some information that cannot.

52. Workshop on Relative Gain Array: Problem 4 You would like to evaluate the steady-state RGA for a process, but the feedback controllers must remain in automatic status. How can you obtain the needed data? Explain a procedure that might yield the needed information. Discuss the practicality of the approach.

53. Workshop on Relative Gain Array: Problem 5 TC LC FC VC SP = 95% open To the SP of the feed flow CV to the VC controller MV from the VC controller MV from the TC controller Determine the relative gain array for the TC and VC controllers. Discuss the behavior of this design and generalize the conclusions.

55. RGA WORKSHOP. PROBLEM 6 (continued)