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Process Control: Designing Process and Control Systems for Dynamic Performance Chapter 20. Multiloop Control – Relative Gain Analysis Copyright © Thomas Marlin 2013 The copyright holder provides a royalty-free license for use of this material at non-profit educational institutions 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. 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. PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA Let’s start here to build understanding RELATIVE GAIN MEASURE OF INTERACTION The relative gain between MVj and CVi is ij . It given in the following equation. Explain in words. MV1(s) G11(s) + G21(s) + CV1(s) Gd1(s) D(s) G12(s) MV2(s) G22(s) Gd2(s) + + CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed CV2(s) Assumes that loops have an integral mode RELATIVE GAIN MEASURE OF INTERACTION OUTLINE OF THE PRESENTATION 1. DEFINITION OF THE RGA Now, how do we determine the value? 2. EVALUATION OF THE RGA 3. INTERPRETATION OF THE RGA 4. PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA RELATIVE GAIN MEASURE OF INTERACTION 1. The RGA can be calculated from openloop values. CV K MV MV K 1 CV CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed CVi k ij MV j MVi kI ij CV j The relative gain array is the element-by-element product of K with K-1 K K1 T ij kij kI ji RELATIVE GAIN MEASURE OF INTERACTION 1. The RGA can be calculated from openloop values. CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed The relative gain array for a 2x2 system is given in the following equation. 1 11 K12 K 21 1 K11 K 22 What is true for the RGA to have 1’s on diagonal? RELATIVE GAIN MEASURE OF INTERACTION 2. The RGA elements are scale independent. Original units CV1 10 10 MV1 MV CV 9 10 2 2 Changing the units of the CV or the capacity of the valve does not change ij . CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed Modified units CV1 1 10 MV *1 CV * .09 1 MV 2 2 MV or1 10MV1 CV1 10 CV2 or CV2 / 10 9 MV2 9 10 RELATIVE GAIN MEASURE OF INTERACTION 3. The rows and columns of the RGA sum to 1.0. CV1 CV2 CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed MV1 10 9 MV2 9 10 For a 2x2 system, how many elements are independent? RELATIVE GAIN MEASURE OF INTERACTION 4. In some cases, the RGA is very sensitive to small errors in the gains, Kij. CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 1 11 K12 K 21 1 K11 K 22 When is this equation very sensitive to errors in the individual gains? RELATIVE GAIN MEASURE OF INTERACTION 5. We can evaluate the RGA of a system with integrating processes, such as levels. CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed Redefine the output as the derivative of the level; then, calculate as normal. m1 m2 = density L A D = density dL A A m1 m2 Fout dt RELATIVE GAIN MEASURE OF INTERACTION OUTLINE OF THE PRESENTATION 1. DEFINITION OF THE RGA How do we use values to evaluate behavior? 2. EVALUATION OF THE RGA 3. INTERPRETATION OF THE RGA 4. PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA RELATIVE GAIN MEASURE OF INTERACTION CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed MVj CVi ij < 0 In this case, the steady-state gains have different signs depending on the status (auto/manual) of other loops A Solvent A CA A CA0 A B Discuss interaction in this system. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 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! RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 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 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. FR XB IAE = 0.58326 ISE = 0.0041497 XB, Bottoms Lt Key 0.985 0.98 0.975 0 100 200 300 400 0.025 0.02 0.015 0.01 0.005 13.8 9 13.7 8.9 Reflux Flow XD, Distillate Lt Key 0.03 0.99 Reboiled Vapor FRB XD IAE = 0.3338 ISE = 0.0012881 0.995 13.6 13.5 13.4 13.3 0 100 200 300 400 8.8 8.7 8.6 0 100 200 300 Time 400 500 8.5 0 100 200 300 Time 400 500 RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 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? RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 0<ij<1 In this case, the steady-state (ML) gain is larger than the 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 RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed ij= 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. MV1(s) What is generally true when ij= 1 ? G11(s) + G21(s) + CV1(s) Gd1(s) D(s) Does ij= 1 indicate no interaction? G12(s) MV2(s) G22(s) Gd2(s) + + CV2(s) RELATIVE GAIN MEASURE OF INTERACTION ij= 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed Solvent Zero heat of reaction Reactant FS >> FR AC TC Calculate the relative gain. Discuss interaction in this system. RELATIVE GAIN MEASURE OF INTERACTION ij= 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. Diagonal gain matrix k 11 k 22 K 0 0 .. .. .. Lower diagonal gain matrix k 11 k 21 k 22 K .. .. k .. n1 0 .. .. .. .. .. CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed Both give an RGA that is diagonal! 1 0 1 I RG A 1 1 0 1 RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 1<ij In this case, the steady-state (ML) gain is larger than the 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. RELATIVE GAIN MEASURE OF INTERACTION MVj CVi CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed 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? 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. PRELIMINARY CONTROL DESIGN IMPLICATIONS OF RGA Let’s evaluate some design guidelines based on RGA RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #1 Select pairings that do not have any ij<0 CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed • 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? RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #2 Select pairings that do not have any ij=0 CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed • 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? 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 RELATIVE GAIN MEASURE OF INTERACTION Proposed Guideline #3 Select a pairing that has RGA elements as close as possible to ij=1 CVi CVi MV j MV j MVk constant other loops open ij CVi CVi MV j MV j CVk constant other loops closed • 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? For set point response, RGA closer to 1.0 is better FR XD FD XD FRB XB FRB XB RGA = 6.09 RGA = 0.39 IAE = 0.25454 ISE = 0.0004554 IAE = 0.045707 ISE = 8.4564e-005 IAE = 0.059056 ISE = 0.00017124 0.986 0.023 0.986 0.022 0.982 0.98 0.022 50 100 150 0.02 200 0.98 0 SAM = 0.31512 SSM = 0.011905 100 150 200 8.6 50 100 150 200 50 100 Time 150 200 13.8 13.7 8.5 8.46 0 50 100 Time 150 200 50 100 150 200 14 8.48 13.5 0 13.9 13.8 13.7 13.6 13.6 0 0.019 SAM = 0.55128 SSM = 0.017408 8.52 Reflux flow Reboiled vapor 8.7 0 8.54 13.9 8.8 0.02 SAM = 0.10303 SSM = 0.0093095 14 8.9 8.5 50 SAM = 0.28826 SSM = 0.00064734 9 0.021 0.982 0.021 0 0.984 Reboiled vapor 0.984 XB, light key 0.023 XD, light key 0.988 XB, light key 0.024 Reflux flow XD, light key IAE = 0.26687 ISE = 0.00052456 0.988 0 50 100 Time 150 200 13.5 0 50 100 Time 150 200 For set point response, RGA farther from 1.0 is better FR XD FD XD FRB XB FRB XB RGA = 6.09 RGA = 0.39 IAE = 0.14463 ISE = 0.00051677 IAE = 0.45265 ISE = 0.0070806 IAE = 0.32334 ISE = 0.0038309 50 100 150 0.01 0 0 100 150 200 8.6 8.55 50 100 Time 150 200 0.01 0 50 100 150 0 200 14 13.5 8.5 13.5 13.4 13.3 8.4 8.3 8.2 8.1 8 0 50 100 Time 150 200 50 100 150 200 SAM = 4.0285 SSM = 0.6871 8.6 13.1 0 SAM = 0.51504 SSM = 0.011985 13.6 13.2 0 0.95 Reflux flow Reboiled vapor Reflux flow 50 SAM = 0.38988 SSM = 0.0085339 8.65 0.02 0.015 0.005 SAM = 0.21116 SSM = 0.0020517 8.7 8.5 0.97 0.96 0.005 200 XB, light key 0.015 0.98 Reboiled vapor 0 XD, light key XB, light key XD, light key Smaller scale 0.975 0.03 0.025 0.02 0.98 IAE = 0.31352 ISE = 0.0027774 0.99 0.025 13 12.5 12 11.5 0 50 100 Time 150 200 11 0 50 100 Time 150 200 RELATIVE GAIN MEASURE OF INTERACTION The RGA gives useful conclusions from S-S information • 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?”