1. Tier II Worked Examples
Module 5 – Controllability Analysis

2. Tier II Statement of intent
The goal of this tier is to demonstrate various concepts and tools of Controllability Analysis using real examples. Some examples will be given, focusing mainly on Controllability Analysis tools. At the end of Tier II, the student should have a general idea of what is:
Relative Gain Array
Niederlinski Index
Controller design
Design of multivariable controllers
Steady State Decoupling (Singular Value Decomposition)
Module 5 – Controllability Analysis

3. 2.0 Inverse of a matrix
To obtain the Relative Gain Array (RGA) of a transfer function matrix, first of all it is necessary to review how to obtain the inverse of a matrix.
The inverse of a matrix does not exist for all matrices, it exists only if:
The matrix is square, and
Its determinant is not zero (non-singular matrix).
Now given the 3x3 matrix A, it is desired to obtain the inverse of the matrix A.
Module 5 – Controllability Analysis

4. Determinant of matrix A.
The determinant of matrix A is obtained using the cofactors based on any row. In this case the first row is selected.
As it can be seen, the determinant is not equal to zero
Module 5 – Controllability Analysis

5. In a 3x3 matrix there is no problem to calculate the determinant
In a 3x3 matrix there is no problem to calculate the determinant. Nevertheless it is important to note that the second cofactor is being multiplied by the factor of (-1).
This is because the cofactor of each matrix element must be multiplied by the following term:
where i is the row number and j is the column number of the element for which the cofactor is calculated.
It means that for matrix A, the cofactors which are multiplied by (-1) are:
C(12),
C(21),
C(23)
and C(32)
Module 5 – Controllability Analysis

6. ú û ù ê ë é = 3 2 5 4 1 A Matrix of cofactors (C).
Now it is necessary to calculate the cofactors matrix (C) for each element of matrix A. ú û ù ê ë é = 3 2 5 4 1 A
And this way the cofactor matrix has been obtained:
Module 5 – Controllability Analysis

7. The Adjoint of A is the transpose of matrix C.
Now the inverse of A is obtained using the determinant and the adjoint of A.
Check now that by matrix multiplication that the identity matrix is obtained multiplying A and A-1.
Module 5 – Controllability Analysis

8. Determinant of matrix A, using Excel.
As can be seen, the amount of work is extensive, just to calculate the inverse of a matrix!.
Therefore, it will now be shown how to obtain the same inverse matrix A-1 of matrix A now using Excel.
Determinant of matrix A, using Excel.
Fill the numbers of the matrix in an excel sheet, using a cell for each element of the matrix.
Module 5 – Controllability Analysis

9. Calculate the matrix determinant, this will be done using cofactors of the first row.
Each cofactor must be calculated using the appropriate formula as shown in the Excel formula bar. Pay attention on cofactor C(12) because it must be negative, as it has been shown previously. + -
Elements C(11), C(12) and C(13) are the cofactors of each element of the first row, which are elements a11, a12 and a13 according to matrix A:
Module 5 – Controllability Analysis

10. Remember that the cofactor of element a12 must be negative!
Calculate the matrix determinant, using the formula shown on the excel sheet.
Remember that the cofactor of element a12 must be negative!
The determinant of this matrix is not equal to zero. For this reason it is possible to obtain its inverse matrix.
Module 5 – Controllability Analysis

11. Calculate the cofactors of matrix A for the second and third rows.
Now remember, the cofactors for the elements C(21), C(23), and C(32), are:
(select one)
POSITIVE
NEGATIVE
As you can see in the Excel sheet, it is necessary to include the sign for each cofactor, and it arithmetically appears on the cell that calculates the matrix determinant (red one). Do not confuse this sign with the one that has been written in front of the matrix in the excel sheet (blue one). The latter one is just to show the sign of the cofactor.
Module 5 – Controllability Analysis

12. Do the same with the second and third rows of matrix C.
Create the cofactors matrix C placing each cofactor in the corresponding place of the element of which it has been calculated, as the excel formula bar shows.
Do the same with the second and third rows of matrix C.
Module 5 – Controllability Analysis

13. Calculate the adjoint of matrix A, this will be done just transposing the matrix C (matrix of cofactors).
Module 5 – Controllability Analysis

14. And do the same for the remaining elements of matrix A-1
Calculate the inverse of matrix A (A-1). To do this, divide the Adjoint matrix by the determinant of matrix A, already calculated.
And do the same for the remaining elements of matrix A-1
Module 5 – Controllability Analysis

15. And do the same for the rest of elements of matrix A-1
Check by matrix multiplication that matrix A multiplied by A-1 gives the Identity Matrix.
And do the same for the rest of elements of matrix A-1
Module 5 – Controllability Analysis

16. As can be seen, even using excel to obtain the inverse of a matrix is still hard work.
But there exist some functions in Excel that allow you to obtain the inverse of a matrix rapidly. This will be shown next.
Module 5 – Controllability Analysis

17. Determinant of matrix A, using an Excel functions.
Now it will be shown how to use some Excel functions to manipulate matrices.
Determinant of matrix A, using an Excel functions.
Fill the numbers, as before, of the matrix in an Excel sheet, using a cell for each element of the matrix. Select the cell where the determinant is to appear. Select function/insert and choose the function mdeterm.
Module 5 – Controllability Analysis

18. Next, it is necessary to select the range of the matrix data.
This way, the determinant of A is easily calculated.
Module 5 – Controllability Analysis

19. Now the transpose matrix of A will be calculated.
Select the cell where the first element a11T is to appear. Select function/insert and choose the function transpose.
Next, select the range of the matrix, as the next slide shows.
Module 5 – Controllability Analysis

20. Select the range of the matrix, as shown below.
Once the range of the matrix has been selected press OK.
Module 5 – Controllability Analysis

21. No values appear in the cell because it is necessary to introduce the formula as an array. To do that, select the range that the transpose matrix will occupy and press F2. Then, press shift+ctrl+enter. The Excel sheet should look like the one below.
Module 5 – Controllability Analysis

22. Then, select the range of the matrix, as shown on the next slide.
It is possible to obtain directly the inverse of a matrix. Select the cell where the first element a11-1 is to appear. Select function/insert and choose the function minverse.
Then, select the range of the matrix, as shown on the next slide.
Module 5 – Controllability Analysis

23. Select the range of the matrix, as shown below.
Once the range of the matrix has been selected press OK.
Module 5 – Controllability Analysis

24. Again no values appear in the cell because it is necessary to introduce the formula as an array. To do that, select the range that the inverse matrix will occupy and press F2. Next, press shift+ctrl+enter. The Excel sheet must look like the one below.
Module 5 – Controllability Analysis

25. The determinant, transpose and inverse of matrix A can easily be obtained. To verify if the Identity matrix is obtained, the function MMULT of excel is used. Matrix A can be multiplied by Matrix A-1.
Multiplying matrices. Select the cell where the element I11 is to appear. Select function/insert and select the function mmult.
Module 5 – Controllability Analysis

26. Once the ranges of the matrices have been selected, press OK.
Next, select the range of each matrix to be multiplied, as shown below.
Once the ranges of the matrices have been selected, press OK.
Module 5 – Controllability Analysis

27. No values appear in the cell because it is necessary to introduce the formula as an array. To do that, select the range that the identity matrix will occupy and press F2. Next, press shift+ctrl+enter. The Excel sheet should look like the one below.
Module 5 – Controllability Analysis

28. Few examples will be covered.
As it was seen, Excel functions are very helpful to obtain the determinant, the transpose and the inverse of a matrix, even for matrix multiplication.
Of course there are several software packages with the availability to work with matrices, but Excel has been chosen because it is available in almost every computer that students have access.
These Excel functions are the main tools that will serve to obtain the Relative Gain Array, as it will be shown shortly.
Few examples will be covered.
Module 5 – Controllability Analysis

29. The size of the array must not exceed 73 rows by 73 columns.
Despite the great help that Excel can provide, some limitations must be specified. These limitations are:
Determinant.
The size of the array must not exceed 73 rows by 73 columns.
Multiplication.
The size of the resulting array must not be equal or greater than a total of cells.
Inverse.
The size of the array must not exceed 52 columns by 52 rows
Module 5 – Controllability Analysis

30. 2.1 Relative Gain Array
2.1.1 Obtain the RGA for the linear model of a distillation column used in separating methanol and water as reported in [1] (see next slide). It is a system with two output variables, two input variables, and one disturbance variable. All variables are defined in terms on deviation variables:
y1= overhead mole fraction methanol
y2= bottoms mole fraction methanol
u1= overhead reflux flowrate
u2= bottoms steam flowrate
d = column feed flowrate
The 2x2 transfer function matrix is:
Module 5 – Controllability Analysis

31. Distillation column used in separating methanol and water
Overhead reflux flow rate (u1)
Overhead mole fraction methanol (y1)
Feed flow rate (d)
Bottoms steam flow rate (u2)
Bottoms mole fraction methanol (y2)
It is easy to identify both manipulated and controlled variables.
Module 5 – Controllability Analysis

32. The steady-state gain matrix is:
Problem description.
From the transfer function matrix G(s), it is possible to obtain the steady-state gain matrix.
The steady-state gain matrix is:
Based on this matrix, the Excel functions seen previously can be used to obtain the RGA as the next slide shows.
Module 5 – Controllability Analysis

33. It is possible to use any Excel sheet shown before, or you can start a new one as the next figure shows. R=
Matrix K
Matrix R
Be careful, because the multiplication of matrices K and R is a multiplication term by term.
Module 5 – Controllability Analysis

34. The RGA has therefore been obtained very easily:
The pairing rules recommend pairing 1-1/2-2, which means to use the overhead reflux flowrate to control the overhead mole fraction of methanol, and the bottoms steam flowrate to control the bottoms mole fraction of methanol.
The final coupling is shown in the next slide.
Module 5 – Controllability Analysis

35. Final coupling suggested for RGA from a distillation column used to separate methanol and water
Overhead reflux flow rate (u1) LC CC
Overhead mole fraction methanol (y1)
Feed flow rate (d)
Bottoms steam flow rate (u2) LC CC
Bottoms mole fraction methanol (y2)
Next, the RGA for a 3x3 system will be calculated.
Module 5 – Controllability Analysis

36. 2.1.2 Obtain the RGA for pilot scale binary distillation column used to separate ethanol and water for which the transfer function matrix is given below [2]. The process variables are (in terms of deviations from their respective steady state values):
Module 5 – Controllability Analysis

37. Distillation column used in separating ethanol and water
Temperature on tray #19 (y3)
Overhead reflux flow rate (u1)
Overhead mole fraction ethanol (y1)
Feed flow rate (d)
Side stream draw-off rate (u2)
Mole fraction of ethanol in the side stream (y2)
Reboiler steam pressure (u3)
Module 5 – Controllability Analysis

38. From the transfer function model, the steady-state gain matrix is:
Problem description.
From the transfer function model, the steady-state gain matrix is:
Matrix K
Matrix R R=
Module 5 – Controllability Analysis

39. The RGA for the 3x3 system has been obtained easily:
Again, remember that the multiplication of matrices K and R is a multiplication term by term.
The RGA for the 3x3 system has been obtained easily:
The pairing rules recommend pairing 1-1/2-2/3-3, which means that the overhead mole fraction of ethanol can be controlled using the overhead reflux flowrate. In the same way, the mole fraction of ethanol in the side stream can be controlled using the side stream draw-off rate and the temperature on tray #19 can be controlled using the reboiler steam pressure.
The next slide shows the final coupling.
Module 5 – Controllability Analysis

40. Final coupling suggested for RGA from a distillation column used in separating ethanol and water
Overhead mole fraction ethanol (y1)
Temperature on tray #19 (y3) CC
Overhead reflux flow rate (u1)
Mole fraction of ethanol in the side stream (y2) TC CC
Feed flow rate (d)
Side stream draw-off rate (u2)
Reboiler steam pressure (u3)
Module 5 – Controllability Analysis

41. 2.2 Niederlinski Index
Up to now the utility of the RGA has been used to find the appropriate pairing for the process variables. Despite the utility of the RGA, sometimes it is necessary to use the RGA with another important tool such as the NIEDERLINSKI INDEX (NI).
The NI is very useful because it allows to identify structurally unstable pairings and as a result to avoid them.
Next the use of NI is shown.
Module 5 – Controllability Analysis

42. 2.2.1 Determine the best pairing using the RGA and the NI for the system with three output variables and three input variables, the steady state matrix is: y3 y1 y2 u3 u1 u2
Problem description.
First of all, since the steady sate matrix is know, the RGA is obtained as before and the next slide shows just the RGA..
Module 5 – Controllability Analysis

43. Then, the RGA obtained is:
Click to
Interchange
Row 2 and 3
According to the RGA, the only feasible pairing has to involve a negative RGA element, so it is possible to interchange rows 2 and 3.
Rows 2 and 3 of RGA have been interchanged. y2 y1 y3 u3 u1 u2
Now the steady state matrix is:
Interchanging the rows was necessary to calculate the Niederlinski Index, because all the pairing elements must be on the diagonal of the K matrix, as next slide shows.
Module 5 – Controllability Analysis

44. Niederlinski Index Niederlinski Index
Now the NI shows that the system is not structurally unstable even pairing a negative element.
Niederlinski Index
Niederlinski Index
1. Obtain the steady state matrix lKl and its determinant.
2. Obtain the diagonal matrix of lKl and its determinant.
3. Obtain the NI ratio:
Module 5 – Controllability Analysis

45. According to these calculations, for the pairing 1-1/2-3/3-2, the system is not unstable, despite the rules of RGA. However if the first loop is opened (y1– u1) or not included in the process model, the resulting subsystem is unstable as will be shown. o
First loop of matrix K open (K): y2 y3 u3 u2 y2 y1 y3 u3 u1 u2
And the NI for matrix K is: o
Reminder: A negative NI indicates that the system is structurally UNSTABLE.
Module 5 – Controllability Analysis

46. 2.3 Nonlinear Systems
Despite that many chemical processes can be adequately represented by linear systems, via linear transfer function models, the majority of chemical processes are inherently nonlinear and sometimes need to use nonlinear models in order to be valid in a wider range of operation.
It is therefore necessary to see how the pairing of input and output variables of nonlinear systems is performed.
Module 5 – Controllability Analysis

47. The same information used for the RGA of steady state systems is used for nonlinear systems. This feature can appear for someone a "disadvantage", but it is precisely this "disadvantage" involving only steady state values that can be used to handle nonlinear systems.
PLEASE !!!!!
Disadvantage or
advantage???
Take a decision
Next, an example shows that according to the steady state values, the pairing of both manipulated and controlled variables is selected.
Module 5 – Controllability Analysis

48. u2= Cold stream temperature
2.3.1 Obtain the RGA for the multivariable system, a stirred mixing tank, consisting in a hot stream and a cold stream which are used to control the liquid level and the tank temperature, and use it to recommend which of the manipulated variables should be used to control the liquid level and the tank temperature. The transfer function matrix is shown below.
y1= Liquid level
y2= Tank temperature
u1= Hot stream flowrate
u2= Cold stream temperature
Where k is a constant (see next slide) and Ac is the cross-section area for the tank.
Module 5 – Controllability Analysis

49. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Problem description.
A diagram of the tank is shown below.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1)
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
F=k(h)½
Next it is necessary to obtain the steady state gain matrix, as next slide shows.
Module 5 – Controllability Analysis

50. Now with the steady-state gain matrix, it is possible to obtain the RGA, using the full matrix method or, since it is a 2x2 matrix, in a more direct way as seen before.
Module 5 – Controllability Analysis

51. Substituting the values of the matrix K:
To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
In Tier 1 has seen that:
and,
Substituting the values of the matrix K:
Module 5 – Controllability Analysis

52. In a similar way it is possible to obtain l12 using:
According to this the value of l12 is:
And the RGA for this systems is:
Module 5 – Controllability Analysis

53. The values given to hot and cold stream are:
Here, it is due to mention that the RGA depends only of values of hot and cold streams, and also for the values of the steady state values involved in the steady-sate gain matrix.
The values given to hot and cold stream are:
TH = 65ºC TC = 15ºC
These values are fixed and just the value of Ts will be changed according to different scenarios.
Module 5 – Controllability Analysis

54. Five different values will given to Ts :
Ts > (TH+TC)/2; Ts = 55ºC
Ts < (TH+TC)/2; Ts = 25ºC
Ts = (TH+TC)/2 ; Ts = 40ºC
Ts = TH: Ts = 65ºC
Ts = TC; Ts = 15ºC
According to this values the system of the stirred mixing tank is:
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
TC = 15ºC
TH = 65ºC
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
Module 5 – Controllability Analysis

55. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 1. Ts > (TH+TC)/2; Ts = 55ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC= 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
F=k(h)½
From the RGA, the suggested pairing is 1-1/2-2.
The physical meaning of this pairing is: since the temperature of cold stream (TC) is farther away from the steady state operating tank temperature, small changes in the cold stream produce noticeable changes in the tank temperature, whereas the temperature of the hot stream (TH) is closer to the operating steady state temperature, it can be used to control the level without causing significant changes in the tank temperature.
Click to
Pairing
1-1/2-2
Module 5 – Controllability Analysis

56. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 2. Ts < (TH+TC)/2; Ts = 25ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-2/2-1.
Again, the physical meaning of this pairing is: since the temperature of the hot stream (TH) is farther away from the steady state operating tank temperature, small changes in the hot stream produce noticeable changes in the tank temperature, whereas the temperature of cold stream (TC) is closer to the operating steady state temperature, it can be used to control the level without causing significant changes in the tank temperature.
Click to
Pairing
1-2/2-1
Module 5 – Controllability Analysis

57. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 3. Ts = (TH+TC)/2; Ts = 40ºC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
Here the values of the RGA are all equal to 0.5 For this reason it is equally bad to pair 1-1/2-2 than 1-2/2-1, because the operating temperature is exactly equidistant from both the cold stream temperature and the hot stream temperature.
A poor control of the process under this undesirable special condition is obtained.
Module 5 – Controllability Analysis

58. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 4. Ts = TH; Ts = 65ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-1/2-2. It is possible to achieve a perfect control of the level tank, without interacting with the temperature, using the hot stream.
Here, the temperature of the hot stream (TH) is the same that the steady state operating tank temperature. For that reason this stream is used to control the level of the tank, whereas the temperature of the cold stream (TC) is used to control the temperature because a small change of cold stream cause significant changes in the tank temperature.
Click to
Pairing
1-1/2-2
Module 5 – Controllability Analysis

59. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 5. Ts = TC; Ts = 15ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-2/2-1. It is possible to achieve a perfect control of the tank level, without affecting the temperature, using the cold stream.
Here, the temperature of the hot stream (TC) is the same than the steady state operating tank temperature. For that reason this stream is used to control the level of the tank, whereas the temperature of hot stream (TH) is used to control the temperature because a small change of hot stream cause significant changes in the tank temperature.
Click to
Pairing
1-2/2-1
Module 5 – Controllability Analysis

60. These five different analyses have demonstrated that the RGA can indeed be used for nonlinear as well as for linear systems.
Pay attention to the fact that the RGA suggests different pairings at different operating conditions. This because even that the analysis has been based on approximate linearized models, this property characteristic of the nonlinear systems is not lost.
STEADY STATE
It is as if the nonlinear system was analyzed on different sections, or slides around `fixed` points, or in this case around steady states.
Module 5 – Controllability Analysis

61. 2.4 Non Square Systems
This section discusses another important point about RGA, the selection of variables for Underdefined and Overdefined systems.
To do this task, it is first of all absolutely necessary to manipulate the non square system in order to obtain a square system. This is done according to the type of non square systems.
The objective in non square systems is pairing, as before, the process variables to minimize the interaction between them.
Next slides show how to obtain a square system from a UNDERDEFINED (therefore non square) system.
Module 5 – Controllability Analysis

62. y1= overhead mole fraction ethanol
2.4.1 Obtain the RGA of a pilot scale binary distillation column used to separate ethanol and water for which the transfer function is given below [2]. In addition, consider the side stream draw-off rate set at a fixed amount and it cannot be changed. Use the same process variables, that in Ex
Problem description.
Now , the side stream draw off rate is not a controlled variable, because it is fixed. For this reason the process model is:
y1= overhead mole fraction ethanol
y2= ethanol mole fraction in side stream
u1= overhead reflux flowrate
d = column feed flowrate
u2= reboiler steam pressure
y3= Temperature on Tray #19
Module 5 – Controllability Analysis

63. Distillation column used in separating ethanol and water
Overhead mole fraction ethanol (y1)
Temperature on tray #19 (y3)
Overhead reflux flow rate (u1)
Feed flow rate (d)
Mole fraction of ethanol in the side stream (y2)
Reboiler steam pressure (u2)
Module 5 – Controllability Analysis

64. It is impossible to control all three output (ys) variables with only two input variables (us).
For that reason it is necessary to select the two most important variables to be controlled, in this case the variables selected have been y1 and y3, as next diagram shows.
Overhead mole fraction ethanol (y1)
Temperature on tray #19 (y3)
Overhead reflux flow rate (u1)
Here the side stream mole fraction of ethanol is taken as the less important of the output variables.
Mole fraction of ethanol in the side stream (y2)
Feed flow rate (d)
Reboiler steam pressure (u2)
Module 5 – Controllability Analysis

65. y1= overhead mole fraction ethanol u1= overhead reflux flowrate
Once it has been decided to leave the control of the side stream composition out of control scheme, the control model is now:
y1= overhead mole fraction ethanol
u1= overhead reflux flowrate
y3= Temperature on Tray #19
u2= reboiler steam pressure
This is a square (modified) subsystem. Therefore, now it is possible to perform the RGA analysis and also to obtain the additional relation:
Module 5 – Controllability Analysis

66. According to these values of RGA, a 1-1/2-2 pairing is recommended.
From the subsystem, the steady state gain matrix and the RGA obtained is:
According to these values of RGA, a 1-1/2-2 pairing is recommended.
It means to use the overhead reflux (u1) to control the overhead composition (y1), and use the reboiler steam pressure (u2) to control Tray #19 temperature (y2). This makes sense.
Module 5 – Controllability Analysis

67. It must notice that according to the relation:
The side stream composition will drift according to the values of the overhead reflux (u1) and the reboiler steam pressure (u2).
This is the nature of UNDERDEFINED systems. The previous system showed that it is only possible to achieve arbitrarily good control of two [overhead mole fraction ethanol (y1) and temperature on Tray #19 (y3)] of the three output variables and accept the drift of the third one (composition on the side stream).
The strategy to work with an UNDERDEFINED system is to choose a square subsystem by dropping off the excess number of output variables on the basis of economic importance; the subsequent analysis is the same as for square systems.
Module 5 – Controllability Analysis

68. Next will be show how to deal with Overdefined systems.
And this is the real challenge of non square systems, so you must put all your attention…
…and follow the instructions given in the next example.
Module 5 – Controllability Analysis

69. 2.4.2 According to a certain system with two outputs (y1 and y2) to be controlled using two of three available inputs (u1, u2, and u3), which loop pairing is expected to give the best control?. Through pulse testing, the following transfer function model was obtained.
Problem description.
This is a 2x3 system, this implies that only two of the three candidate input variables will be used for control, while the third input variable will have to be set at a fixed value and will therefore be redundant.
To determine which variables should be active and which ones should be redundant, first of all possible 2x2 subsystems must be obtained.
Module 5 – Controllability Analysis

70. Subsystem 1. Utilizing u1 and u2 for control:
Where: u are the number of input variables and n the number of output variables.
According to the previous system: u=3 and n=2
Subsystem 1. Utilizing u1 and u2 for control:
Module 5 – Controllability Analysis

71. From the subsystem 1, the steady state matrix and the RGA are:
Subsystem 2. Utilizing u1 and u2 for control:
Module 5 – Controllability Analysis

72. From the subsystem 2, the steady state matrix and the RGA are:
Subsystem 3. Utilizing u2 and u3 for control:
Module 5 – Controllability Analysis

73. From the subsystem 3, the steady state matrix and the RGA are:
Next slide shows the three RGA values obtained for each subsystem.
Module 5 – Controllability Analysis

74. RGA for subsystems 1 to 3.
According to these values of RGA for each subsystem, the best possible control is the subsystem 1, because it is closest to the ideal situation; it is somewhat better than subsystem 2 and far superior than subsystem 3.
This subsystem involves pairing u1 with y1 and y2 with u2, and this also implies that u3 is to be redundant.
Module 5 – Controllability Analysis

75. 2.5 Factors Influencing the Loop Pairing
As seen in TIER I, there are some factors that affect how the variables are paired. Some of those are:
Constraints in the input variable,
Time delay,
Inverse response,
Slow dynamics in the best RGA paring,
Timescale decoupling of loop dynamics
Next slides show how to pair the process variables according to these factors.
Module 5 – Controllability Analysis

76. u2= Cold stream temperature
2.5.1 Take again Ex , but now, an in-tank heater was added to the stirred mixing tank to control the temperature with the heater power Q. Obtain the RGA for this system if Ts=(TH+TC)/2. The transfer function is:
y1= Liquid level
y2= Tank temperature
u1= Hot stream flowrate
u2= Cold stream temperature
u3= Heater power
Where k is the same constant as in Ex and Ac is the cross-section area for the tank.
Module 5 – Controllability Analysis

77. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Problem description.
A diagram of the tank with the Heater is show below.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1) h
HEAT POWER (u3)
TANK LIQUID LEVEL (y1)
F=k(h)½
Output Flow rate (F, T)
Now it is an overdetermined system with more than one subsystem to pair. First the RGA for each subsystem will be obtained.
Module 5 – Controllability Analysis

78. In a similar way as Ex. 2.4.2, there are three different subsystems:
Subsystem 1. Utilizing u1 (Hot Stream) and u2 (Cold stream) for control.
The steady state gain for this subsystem is the same that ex :
And the RGA is the same as obtained in ex :
Module 5 – Controllability Analysis

79. Subsystem 2. Utilizing u1 (Hot Stream) and u3 (Heater) for control.
The transfer function matrix for this subsystem is:
And the steady state gain matrix is:
Module 5 – Controllability Analysis

80. To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
Again:
and,
Substituting the values of the matrix K of this subsystem:
Therefore l11 is:
Module 5 – Controllability Analysis

81. Next the RGA for subsystem 3, will be obtained.
And since, l12:
Finally the value of l12 is:
And the RGA for subsystem 2 is :
Next the RGA for subsystem 3, will be obtained.
Module 5 – Controllability Analysis

82. Subsystem 3. Utilizing u2 (Cold Stream) y u3 (Heater) for control.
The transfer function matrix for this subsystem is:
And the steady state gain matrix is:
Module 5 – Controllability Analysis

83. To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
Again:
and,
Substituting the values of the matrix K of this subsystem:
Therefore l11 is:
Module 5 – Controllability Analysis

84. Finally the value of l12 is:
And since, l12:
Finally the value of l12 is:
And the RGA for subsystem 3 is :
You must noted that the RGA for subsystem 2 and 3 is the same and both are independent of Ts.
Module 5 – Controllability Analysis

85. RGA for subsystems 1 to 3.
Taking the case where Ts=(TH+TC)/2, the RGA for each subsystem is:
Subsystem 1.
Subsystem 2.
Subsystem 3.
Again, note that the RGA for subsystem 1 was obtained in Ex Subsystems 2 and 3 are the same and both are independent of Ts.
Module 5 – Controllability Analysis

86. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
According to this analysis, the pairing in subsystem 2 involves to use the Hot stream temperature (u1) to control the liquid level (y1) and use the in tank heater (u3) to control the tank temperature (y2):
Subsystem 2.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
HOT STREAM FLOWRATE (u1)
DISTURBANCE (Td, Fd)
Subsystem 3.
LLC h
TANK LIQUID LEVEL (y1)
HEAT POWER (u3)
F=k(h)½ TC
Output Flow rate (F, T)
Click to
Pairing
Subsystem 2
Module 5 – Controllability Analysis

87. For subsystem 3 the pairing involves to use the Cold stream temperature (u2) to control the liquid level (y1) and use the in-tank heater (u3) to control the tank temperature (y2):
Subsystem 3.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
HOT STREAM FLOWRATE (u1)
DISTURBANCE (Td, Fd)
Subsystem 3.
LLC
TANK LIQUID LEVEL (y1) h
HEAT POWER (u3)
F=k(h)½ TC
Output Flow rate (F, T)
Click to
Pairing
Subsystem 3
Module 5 – Controllability Analysis

88. But both pairings of these subsystems can become an undesirable pairing control as it will be discussed next.
If the in-tank heater can barely achieve the steady state, Ts, at maximum power, there is a major problem.
Thus, this subsystem would not be desirable for the regulatory temperature control because, following variations of the other process variable (hot or cold stream), the IN-TANK HEATER has no more power to supply (or extract) heat to keep the new steady state temperature.
Module 5 – Controllability Analysis

89. Now to overcome the power limitation, a much larger heater is installed in the tank, but as a consequence of this, there is a , between the control signal and the actual power delivery.
VERY LARGE TIME DELAY
And because of this sluggish closed-loop response in the heater, the best choice for pairing the process variables could be the poor RGA of subsystem 1.
Next will be show another factor to considerer in the loop pairing of process variables.
Module 5 – Controllability Analysis

90. 2.5.2 Now considering a system with a transfer function given as below, obtain the RGA for this system and analyze a unit set point change in (y1) and a diagonal PI controller (Kc1= 4, I1=0.5; Kc2=-4, I2=0.3) using the resulting pairing. t t
Problem description.
First of all it is necessary to obtain the steady state gain matrix, as it is shows below.
Module 5 – Controllability Analysis

91. Now, the RGA obtained as before is:
According to the RGA, the recommended pairing is y1-u1 and y2-u2.
Next step is to analyze a change in the set point.
Module 5 – Controllability Analysis

92. Since the dynamic simulation of the analyzed system is beyond the scope of this module, only the result of the change in the set point will be display.
Module 5 – Controllability Analysis

93. As mentioned in last slide, the next graphic shows the closed loop response for a unit set point change in y1 using the pairing suggested for the RGA and a diagonal PI controller.
Pairing 1-1/2-2 y1
Set point y1
The performance of this pairing is not too bad considering that the open loop time constants on the diagonal are 10 minutes
Despite this “not too bad” performance, the inverse pairing will be analyzed for the same set point change. y2
Set point y2
Inverse loop pairing involves to take the value of l=0.2 in the pairing, but it has been mentioned as a situation to avoid !!!!!!!!.
Module 5 – Controllability Analysis

94. Different pairing also implies to use a different PI controller, for that reason the inverse pairing analysis of a unit set point change in (y1), the new diagonal PI controller is (Kc1= 10, I1=0.3; Kc2= 20, I2=0.3). t t
Pairing 1-2/2-1
Set point y1 y1
The reason is that the control loops are able to respond so rapidly that the interactions that appear more slowly are easily dealt with.
The performance in this case is dramatically better than the recommended pairing by the RGA, because the open loop time constants on the diagonal are only 1 minute.
Set point y2 y2
Finally in this example, the best loop pairing was obtained using the inverse pairing, than the suggested by the RGA.
Module 5 – Controllability Analysis

95. You should not fell like any of this, because…
After this example, do you fell like this ?...
You should not fell like any of this, because…
The purpose of this example is not to confuse you about how to select a loop pairing, the purpose is
to show you that RGA provides only a guideline
to steady state interactions, for that reason, all other engineering considerations must be used together in choosing the loop pairing.
Module 5 – Controllability Analysis

96. 2.3 Nonlinear Systems
Despite that many chemical processes can be adequately represented by linear systems, via linear transfer function models, the majority of chemical processes are inherently nonlinear and sometimes need to use nonlinear models in order to be valid in a wider range of operation.
It is therefore necessary to see how the pairing of input and output variables of nonlinear systems is performed.
Module 5 – Controllability Analysis

97. The same information used for the RGA of steady state systems is used for nonlinear systems. This feature can appear for someone a "disadvantage", but it is precisely this "disadvantage" involving only steady state values that can be used to handle nonlinear systems.
PLEASE !!!!!
Disadvantage or
advantage???
Take a decision
Next, an example shows that according to the steady state values, the pairing of both manipulated and controlled variables is selected.
Module 5 – Controllability Analysis

98. u2= Cold stream temperature
2.3.1 Obtain the RGA for the multivariable system, a stirred mixing tank, consisting in a hot stream and a cold stream which are used to control the liquid level and the tank temperature, and use it to recommend which of the manipulated variables should be used to control the liquid level and the tank temperature. The transfer function matrix is shown below.
y1= Liquid level
y2= Tank temperature
u1= Hot stream flowrate
u2= Cold stream temperature
Where k is a constant (see next slide) and Ac is the cross-section area for the tank.
Module 5 – Controllability Analysis

99. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Problem description.
A diagram of the tank is shown below.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1)
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
F=k(h)½
Next it is necessary to obtain the steady state gain matrix, as next slide shows.
Module 5 – Controllability Analysis

100. Now with the steady-state gain matrix, it is possible to obtain the RGA, using the full matrix method or, since it is a 2x2 matrix, in a more direct way as seen before.
Module 5 – Controllability Analysis

101. Substituting the values of the matrix K:
To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
In Tier 1 has seen that:
and,
Substituting the values of the matrix K:
Module 5 – Controllability Analysis

102. In a similar way it is possible to obtain l12 using:
According to this the value of l12 is:
And the RGA for this systems is:
Module 5 – Controllability Analysis

103. The values given to hot and cold stream are:
Here, it is due to mention that the RGA depends only of values of hot and cold streams, and also for the values of the steady state values involved in the steady-sate gain matrix.
The values given to hot and cold stream are:
TH = 65ºC TC = 15ºC
These values are fixed and just the value of Ts will be changed according to different scenarios.
Module 5 – Controllability Analysis

104. Five different values will given to Ts :
Ts > (TH+TC)/2; Ts = 55ºC
Ts < (TH+TC)/2; Ts = 25ºC
Ts = (TH+TC)/2 ; Ts = 40ºC
Ts = TH: Ts = 65ºC
Ts = TC; Ts = 15ºC
According to this values the system of the stirred mixing tank is:
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
TC = 15ºC
TH = 65ºC
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
Module 5 – Controllability Analysis

105. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 1. Ts > (TH+TC)/2; Ts = 55ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC= 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
F=k(h)½
From the RGA, the suggested pairing is 1-1/2-2.
The physical meaning of this pairing is: since the temperature of cold stream (TC) is farther away from the steady state operating tank temperature, small changes in the cold stream produce noticeable changes in the tank temperature, whereas the temperature of the hot stream (TH) is closer to the operating steady state temperature, it can be used to control the level without causing significant changes in the tank temperature.
Click to
Pairing
1-1/2-2
Module 5 – Controllability Analysis

106. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 2. Ts < (TH+TC)/2; Ts = 25ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-2/2-1.
Again, the physical meaning of this pairing is: since the temperature of the hot stream (TH) is farther away from the steady state operating tank temperature, small changes in the hot stream produce noticeable changes in the tank temperature, whereas the temperature of cold stream (TC) is closer to the operating steady state temperature, it can be used to control the level without causing significant changes in the tank temperature.
Click to
Pairing
1-2/2-1
Module 5 – Controllability Analysis

107. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 3. Ts = (TH+TC)/2; Ts = 40ºC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1) h
Output Flow rate (F, T)
Here the values of the RGA are all equal to 0.5 For this reason it is equally bad to pair 1-1/2-2 than 1-2/2-1, because the operating temperature is exactly equidistant from both the cold stream temperature and the hot stream temperature.
A poor control of the process under this undesirable special condition is obtained.
Module 5 – Controllability Analysis

108. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 4. Ts = TH; Ts = 65ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-1/2-2. It is possible to achieve a perfect control of the level tank, without interacting with the temperature, using the hot stream.
Here, the temperature of the hot stream (TH) is the same that the steady state operating tank temperature. For that reason this stream is used to control the level of the tank, whereas the temperature of the cold stream (TC) is used to control the temperature because a small change of cold stream cause significant changes in the tank temperature.
Click to
Pairing
1-1/2-2
Module 5 – Controllability Analysis

109. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Case 5. Ts = TC; Ts = 15ºC TC
TANK TEMPERATURE (y2)
COLD STREAM FLOWRATE (u2)
TC = 15ºC
DISTURBANCE (Td,Fd)
TH = 65ºC
HOT STREAM FLOWRATE (u1)
F=k(h)½
TANK LIQUID LEVEL (y1)
LLC h
Output Flow rate (F, T)
From the RGA, the suggested pairing is 1-2/2-1. It is possible to achieve a perfect control of the tank level, without affecting the temperature, using the cold stream.
Here, the temperature of the hot stream (TC) is the same than the steady state operating tank temperature. For that reason this stream is used to control the level of the tank, whereas the temperature of hot stream (TH) is used to control the temperature because a small change of hot stream cause significant changes in the tank temperature.
Click to
Pairing
1-2/2-1
Module 5 – Controllability Analysis

110. These five different analyses have demonstrated that the RGA can indeed be used for nonlinear as well as for linear systems.
Pay attention to the fact that the RGA suggests different pairings at different operating conditions. This because even that the analysis has been based on approximate linearized models, this property characteristic of the nonlinear systems is not lost.
STEADY STATE
It is as if the nonlinear system was analyzed on different sections, or slides around `fixed` points, or in this case around steady states.
Module 5 – Controllability Analysis

111. 2.4 Non Square Systems
This section discusses another important point about RGA, the selection of variables for Underdefined and Overdefined systems.
To do this task, it is first of all absolutely necessary to manipulate the non square system in order to obtain a square system. This is done according to the type of non square systems.
The objective in non square systems is pairing, as before, the process variables to minimize the interaction between them.
Next slides show how to obtain a square system from a UNDERDEFINED (therefore non square) system.
Module 5 – Controllability Analysis

112. y1= overhead mole fraction ethanol
2.4.1 Obtain the RGA of a pilot scale binary distillation column used to separate ethanol and water for which the transfer function is given below [2]. In addition, consider the side stream draw-off rate set at a fixed amount and it cannot be changed. Use the same process variables, that in Ex
Problem description.
Now , the side stream draw off rate is not a controlled variable, because it is fixed. For this reason the process model is:
y1= overhead mole fraction ethanol
y2= ethanol mole fraction in side stream
u1= overhead reflux flowrate
d = column feed flowrate
u2= reboiler steam pressure
y3= Temperature on Tray #19
Module 5 – Controllability Analysis

113. Distillation column used in separating ethanol and water
Overhead mole fraction ethanol (y1)
Temperature on tray #19 (y3)
Overhead reflux flow rate (u1)
Feed flow rate (d)
Mole fraction of ethanol in the side stream (y2)
Reboiler steam pressure (u2)
Module 5 – Controllability Analysis

114. It is impossible to control all three output (ys) variables with only two input variables (us).
For that reason it is necessary to select the two most important variables to be controlled, in this case the variables selected have been y1 and y3, as next diagram shows.
Overhead mole fraction ethanol (y1)
Temperature on tray #19 (y3)
Overhead reflux flow rate (u1)
Here the side stream mole fraction of ethanol is taken as the less important of the output variables.
Mole fraction of ethanol in the side stream (y2)
Feed flow rate (d)
Reboiler steam pressure (u2)
Module 5 – Controllability Analysis

115. y1= overhead mole fraction ethanol u1= overhead reflux flowrate
Once it has been decided to leave the control of the side stream composition out of control scheme, the control model is now:
y1= overhead mole fraction ethanol
u1= overhead reflux flowrate
y3= Temperature on Tray #19
u2= reboiler steam pressure
This is a square (modified) subsystem. Therefore, now it is possible to perform the RGA analysis and also to obtain the additional relation:
Module 5 – Controllability Analysis

116. According to these values of RGA, a 1-1/2-2 pairing is recommended.
From the subsystem, the steady state gain matrix and the RGA obtained is:
According to these values of RGA, a 1-1/2-2 pairing is recommended.
It means to use the overhead reflux (u1) to control the overhead composition (y1), and use the reboiler steam pressure (u2) to control Tray #19 temperature (y2). This makes sense.
Module 5 – Controllability Analysis

117. It must notice that according to the relation:
The side stream composition will drift according to the values of the overhead reflux (u1) and the reboiler steam pressure (u2).
This is the nature of UNDERDEFINED systems. The previous system showed that it is only possible to achieve arbitrarily good control of two [overhead mole fraction ethanol (y1) and temperature on Tray #19 (y3)] of the three output variables and accept the drift of the third one (composition on the side stream).
The strategy to work with an UNDERDEFINED system is to choose a square subsystem by dropping off the excess number of output variables on the basis of economic importance; the subsequent analysis is the same as for square systems.
Module 5 – Controllability Analysis

118. Next will be show how to deal with Overdefined systems.
And this is the real challenge of non square systems, so you must put all your attention…
…and follow the instructions given in the next example.
Module 5 – Controllability Analysis

119. 2.4.2 According to a certain system with two outputs (y1 and y2) to be controlled using two of three available inputs (u1, u2, and u3), which loop pairing is expected to give the best control?. Through pulse testing, the following transfer function model was obtained.
Problem description.
This is a 2x3 system, this implies that only two of the three candidate input variables will be used for control, while the third input variable will have to be set at a fixed value and will therefore be redundant.
To determine which variables should be active and which ones should be redundant, first of all possible 2x2 subsystems must be obtained.
Module 5 – Controllability Analysis

120. Subsystem 1. Utilizing u1 and u2 for control:
Where: u are the number of input variables and n the number of output variables.
According to the previous system: u=3 and n=2
Subsystem 1. Utilizing u1 and u2 for control:
Module 5 – Controllability Analysis

121. From the subsystem 1, the steady state matrix and the RGA are:
Subsystem 2. Utilizing u1 and u2 for control:
Module 5 – Controllability Analysis

122. From the subsystem 2, the steady state matrix and the RGA are:
Subsystem 3. Utilizing u2 and u3 for control:
Module 5 – Controllability Analysis

123. From the subsystem 3, the steady state matrix and the RGA are:
Next slide shows the three RGA values obtained for each subsystem.
Module 5 – Controllability Analysis

124. RGA for subsystems 1 to 3.
According to these values of RGA for each subsystem, the best possible control is the subsystem 1, because it is closest to the ideal situation; it is somewhat better than subsystem 2 and far superior than subsystem 3.
This subsystem involves pairing u1 with y1 and y2 with u2, and this also implies that u3 is to be redundant.
Module 5 – Controllability Analysis

125. 2.5 Factors Influencing the Loop Pairing
As seen in TIER I, there are some factors that affect how the variables are paired. Some of those are:
Constraints in the input variable,
Time delay,
Inverse response,
Slow dynamics in the best RGA paring,
Timescale decoupling of loop dynamics
Next slides show how to pair the process variables according to these factors.
Module 5 – Controllability Analysis

126. u2= Cold stream temperature
2.5.1 Take again Ex , but now, an in-tank heater was added to the stirred mixing tank to control the temperature with the heater power Q. Obtain the RGA for this system if Ts=(TH+TC)/2. The transfer function is:
y1= Liquid level
y2= Tank temperature
u1= Hot stream flowrate
u2= Cold stream temperature
u3= Heater power
Where k is the same constant as in Ex and Ac is the cross-section area for the tank.
Module 5 – Controllability Analysis

127. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
Problem description.
A diagram of the tank with the Heater is show below.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
DISTURBANCE (Td, Fd)
HOT STREAM FLOWRATE (u1) h
HEAT POWER (u3)
TANK LIQUID LEVEL (y1)
F=k(h)½
Output Flow rate (F, T)
Now it is an overdetermined system with more than one subsystem to pair. First the RGA for each subsystem will be obtained.
Module 5 – Controllability Analysis

128. In a similar way as Ex. 2.4.2, there are three different subsystems:
Subsystem 1. Utilizing u1 (Hot Stream) and u2 (Cold stream) for control.
The steady state gain for this subsystem is the same that ex :
And the RGA is the same as obtained in ex :
Module 5 – Controllability Analysis

129. Subsystem 2. Utilizing u1 (Hot Stream) and u3 (Heater) for control.
The transfer function matrix for this subsystem is:
And the steady state gain matrix is:
Module 5 – Controllability Analysis

130. To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
Again:
and,
Substituting the values of the matrix K of this subsystem:
Therefore l11 is:
Module 5 – Controllability Analysis

131. Next the RGA for subsystem 3, will be obtained.
And since, l12:
Finally the value of l12 is:
And the RGA for subsystem 2 is :
Next the RGA for subsystem 3, will be obtained.
Module 5 – Controllability Analysis

132. Subsystem 3. Utilizing u2 (Cold Stream) y u3 (Heater) for control.
The transfer function matrix for this subsystem is:
And the steady state gain matrix is:
Module 5 – Controllability Analysis

133. To calculate the RGA of this 2x2 system it is possible to obtain l11 using the First Principles.
Again:
and,
Substituting the values of the matrix K of this subsystem:
Therefore l11 is:
Module 5 – Controllability Analysis

134. Finally the value of l12 is:
And since, l12:
Finally the value of l12 is:
And the RGA for subsystem 3 is :
You must noted that the RGA for subsystem 2 and 3 is the same and both are independent of Ts.
Module 5 – Controllability Analysis

135. RGA for subsystems 1 to 3.
Taking the case where Ts=(TH+TC)/2, the RGA for each subsystem is:
Subsystem 1.
Subsystem 2.
Subsystem 3.
Again, note that the RGA for subsystem 1 was obtained in Ex Subsystems 2 and 3 are the same and both are independent of Ts.
Module 5 – Controllability Analysis

136. COLD STREAM FLOWRATE (u2) HOT STREAM FLOWRATE (u1)
According to this analysis, the pairing in subsystem 2 involves to use the Hot stream temperature (u1) to control the liquid level (y1) and use the in tank heater (u3) to control the tank temperature (y2):
Subsystem 2.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
HOT STREAM FLOWRATE (u1)
DISTURBANCE (Td, Fd)
Subsystem 3.
LLC h
TANK LIQUID LEVEL (y1)
HEAT POWER (u3)
F=k(h)½ TC
Output Flow rate (F, T)
Click to
Pairing
Subsystem 2
Module 5 – Controllability Analysis

137. For subsystem 3 the pairing involves to use the Cold stream temperature (u2) to control the liquid level (y1) and use the in-tank heater (u3) to control the tank temperature (y2):
Subsystem 3.
COLD STREAM FLOWRATE (u2)
TANK TEMPERATURE (y2)
HOT STREAM FLOWRATE (u1)
DISTURBANCE (Td, Fd)
Subsystem 3.
LLC
TANK LIQUID LEVEL (y1) h
HEAT POWER (u3)
F=k(h)½ TC
Output Flow rate (F, T)
Click to
Pairing
Subsystem 3
Module 5 – Controllability Analysis

138. But both pairings of these subsystems can become an undesirable pairing control as it will be discussed next.
If the in-tank heater can barely achieve the steady state, Ts, at maximum power, there is a major problem.
Thus, this subsystem would not be desirable for the regulatory temperature control because, following variations of the other process variable (hot or cold stream), the IN-TANK HEATER has no more power to supply (or extract) heat to keep the new steady state temperature.
Module 5 – Controllability Analysis

139. Now to overcome the power limitation, a much larger heater is installed in the tank, but as a consequence of this, there is a , between the control signal and the actual power delivery.
VERY LARGE TIME DELAY
And because of this sluggish closed-loop response in the heater, the best choice for pairing the process variables could be the poor RGA of subsystem 1.
Next will be show another factor to considerer in the loop pairing of process variables.
Module 5 – Controllability Analysis

140. 2.5.2 Now considering a system with a transfer function given as below, obtain the RGA for this system and analyze a unit set point change in (y1) and a diagonal PI controller (Kc1= 4, I1=0.5; Kc2=-4, I2=0.3) using the resulting pairing. t t
Problem description.
First of all it is necessary to obtain the steady state gain matrix, as it is shows below.
Module 5 – Controllability Analysis

141. Now, the RGA obtained as before is:
According to the RGA, the recommended pairing is y1-u1 and y2-u2.
Next step is to analyze a change in the set point.
Module 5 – Controllability Analysis

142. Since the dynamic simulation of the analyzed system is beyond the scope of this module, only the result of the change in the set point will be display.
Module 5 – Controllability Analysis

143. As mentioned in last slide, the next graphic shows the closed loop response for a unit set point change in y1 using the pairing suggested for the RGA and a diagonal PI controller.
Pairing 1-1/2-2 y1
Set point y1
The performance of this pairing is not too bad considering that the open loop time constants on the diagonal are 10 minutes
Despite this “not too bad” performance, the inverse pairing will be analyzed for the same set point change. y2
Set point y2
Inverse loop pairing involves to take the value of l=0.2 in the pairing, but it has been mentioned as a situation to avoid !!!!!!!!.
Module 5 – Controllability Analysis

144. Different pairing also implies to use a different PI controller, for that reason the inverse pairing analysis of a unit set point change in (y1), the new diagonal PI controller is (Kc1= 10, I1=0.3; Kc2= 20, I2=0.3). t t
Pairing 1-2/2-1
Set point y1 y1
The reason is that the control loops are able to respond so rapidly that the interactions that appear more slowly are easily dealt with.
The performance in this case is dramatically better than the recommended pairing by the RGA, because the open loop time constants on the diagonal are only 1 minute.
Set point y2 y2
Finally in this example, the best loop pairing was obtained using the inverse pairing, than the suggested by the RGA.
Module 5 – Controllability Analysis

145. You should not fell like any of this, because…
After this example, do you fell like this ?...
You should not fell like any of this, because…
The purpose of this example is not to confuse you about how to select a loop pairing, the purpose is
to show you that RGA provides only a guideline
to steady state interactions, for that reason, all other engineering considerations must be used together in choosing the loop pairing.
Module 5 – Controllability Analysis