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RELATIVE GAIN MEASURE OF INTERACTION

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 ?

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RELATIVE GAIN MEASURE OF INTERACTION

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  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 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 CONTROLDESIGN IMPLICATIONS OF RGA Let’s start here to build understanding

  4. G11(s) MV1(s) CV1(s) + + G21(s) Gd1(s) D(s) G12(s) Gd2(s) G22(s) MV2(s) CV2(s) + + RELATIVE GAIN MEASURE OF INTERACTION The relative gain between MVj and CVi is ij . It is defined in the following equation. Explain in words. What have we assumed about the other controllers?

  5. 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 CONTROLDESIGN IMPLICATIONS OF RGA Now, how do we determine the value?

  6. The relative gain array is the element-by-element product of K with K-1. ( = product of ij elements, not normal matrix multiplication) RELATIVE GAIN MEASURE OF INTERACTION 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. 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 ? Original units Modified units 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, Kij. 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, Kij. We must perform a thorough study to ensure that numerical derivatives are sufficiently accurate! From McAvpy, 1983 The x must be sufficiently small (be careful about roundoff).

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

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

  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 gainRGA ??

  17. 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 CONTROLDESIGN IMPLICATIONS OF RGA How do we use values to evaluate behavior?

  18. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi ij < 0 In this case, the steady-state gains have different signs depending on the status (auto/manual) of the other loops. A CA0 CSTR with A  B Discuss interaction and RGA in this system. Solvent A CA A

  19. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi 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 MVj CVi 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. IAE = 0.3338 ISE = 0.0012881 IAE = 0.58326 ISE = 0.0041497 0.995 0.03 0.025 0.99 0.02 XD, Distillate Lt Key 0.985 XB, Bottoms Lt Key 0.015 0.98 0.01 0.975 0.005 0 100 200 300 400 0 100 200 300 400 13.8 9 13.7 8.9 13.6 8.8 Reflux Flow Reboiled Vapor 13.5 8.7 13.4 8.6 13.3 8.5 0 100 200 300 400 500 0 100 200 300 400 500 Time Time 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  XB FV  XD XD FR FV XB

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

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

  24. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi 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. G11(s) MV1(s) CV1(s) + + G21(s) Gd1(s) D(s) G12(s) Gd2(s) G22(s) MV2(s) CV2(s) + + RELATIVE GAIN MEASURE OF INTERACTION MVj CVi ij= 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. What is generally true when ij= 1 ? Does ij= 1 indicate no interaction?

  26. Solvent CSTR with zero heat of reaction Reactant FS >> FR AC TC RELATIVE GAIN MEASURE OF INTERACTION ij= 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. 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. Both give an RGA that is diagonal! 0 Diagonal gain matrix Diagonal gain matrix 0 0 0 0 Lower diagonal gain matrix

  28. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi 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 athe case of a 2x2 system paired on ij = 10.

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

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

  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 CONTROLDESIGN IMPLICATIONS OF RGA Let’s extend the concept

  32. RELATIVE GAIN MEASURE OF INTERACTION The relative gain between MVj and CVi 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. m1 m m2  = density L A D = density 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.)

  34. m1 m m2  = density L A D = density 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.

  35. G11(s) MV1(s) CV1(s) + + G21(s) Gd1(s) D(s) G12(s) Gd2(s) G22(s) MV2(s) CV2(s) + + RELATIVE GAIN MEASURE OF INTERACTION A frequency-dependent RGA can be calculated using the transfer functions in place of the steady-state gains. We can evaluate the RGA of dynamics processes FR  XD FRB  XB

  36. 0 0 10 10 amplitude, XD(jw)/FV(jw) amplitude, XD(jw)/FR(jw) -5 -5 10 10 -4 -2 0 2 -4 -2 0 2 10 10 10 10 10 10 10 10 frequency, radians/min 0 0 10 10 -2 -2 amplitude, XB(jw)/FV(jw) 10 10 amplitude, XB(jw)/FR(jw) -4 -4 10 10 -4 -2 0 2 -4 -2 0 2 10 10 10 10 10 10 10 10 Bode plots of the individual transfer functions for a distillation tower

  37. frequency dependent RGA for distillation tower 1 10 0 10 amplitude ratio -1 10 -3 -2 -1 0 1 2 10 10 10 10 10 10 frequency, rad/time 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 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 CONTROLDESIGN IMPLICATIONS OF RGA Let’s evaluate some design guidelines based on 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 RGA and INTEGRITY • 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

  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. FD  XD FRB  XB FR  XD FRB  XB IAE = 0.045707 ISE = 8.4564e-005 IAE = 0.25454 ISE = 0.0004554 IAE = 0.059056 ISE = 0.00017124 IAE = 0.26687 ISE = 0.00052456 0.023 0.024 0.988 0.988 0.986 0.022 0.986 0.023 XD, light key XB, light key 0.984 0.021 0.984 0.022 XD, light key XB, light key 0.982 0.02 0.982 0.021 0.019 0.98 0.02 0.98 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 SAM = 0.10303 SSM = 0.0093095 SAM = 0.55128 SSM = 0.017408 SAM = 0.31512 SSM = 0.011905 SAM = 0.28826 SSM = 0.00064734 8.54 14 9 14 13.9 8.9 13.9 8.52 13.8 8.8 13.8 Reflux flow 8.5 Reboiled vapor Reflux flow Reboiled vapor 13.7 8.7 13.7 8.48 13.6 8.6 13.6 8.46 13.5 8.5 13.5 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Time Time Time Time For set point response, RGA closer to 1.0 is better RGA = 6.09 RGA = 0.39

  45. FD  XD FRB  XB FR  XD FRB  XB IAE = 0.45265 ISE = 0.0070806 IAE = 0.31352 ISE = 0.0027774 IAE = 0.14463 ISE = 0.00051677 IAE = 0.32334 ISE = 0.0038309 0.025 0.99 0.03 0.025 0.02 0.98 0.02 0.98 0.015 XD, light key XB, light key 0.97 0.015 XD, light key XB, light key 0.01 0.01 0.96 0.975 0.005 0.005 0.95 0 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 SAM = 0.51504 SSM = 0.011985 SAM = 4.0285 SSM = 0.6871 SAM = 0.21116 SSM = 0.0020517 SAM = 0.38988 SSM = 0.0085339 8.6 14 8.7 13.6 8.5 13.5 13.5 8.65 8.4 13 13.4 Reflux flow 8.3 12.5 8.6 Reflux flow Reboiled vapor Reboiled vapor 13.3 8.2 12 8.55 13.2 8.1 11.5 8 11 8.5 13.1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Time Time Time Time For feed composition disturbance response, RGA farther from 1.0 is better RGA = 6.09 RGA = 0.39

  46. RELATIVE GAIN MEASURE OF INTERACTION Using guidelines #1 and #2, the control possibilities for this example process were reduced from 36 to4.

  47. RELATIVE GAIN MEASURE OF INTERACTION The RGA gives useful conclusions from S-S information, but not enough to design process control • 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 Powerful results from limited information! Can we design controls without this information? “Interaction?”

  48. INTERACTION IN FEEDBACK SYSTEMS Workshop on Relative Gain Array

  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. A CA0 CSTR with A  B Solvent A CA A 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.

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