Method for model gain matrix modification

Abstract

A method is presented for adjusting the steady-state gains of a multivariable predictive control, planning or optimization model with uncertainty. The user selects a desired matrix relative gain criteria for the predictive model or sub-model. This is used to calculate a base number. Model gains are extracted from the predictive model and the magnitudes are modified to be rounded number powers of the calculated base number.

Description

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a flow diagram illustrating a simple distillation unit having two independent variables and two controlled variables.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A detailed description is demonstrated by an example problem. Consider a predictive model with 2 independent variables and 2 dependent variables. The gain matrix represents the interaction between both independent variables and both dependent variables. Table 1 shows an example of a 2×2 model prediction matrix.

A simple light ends distillation tower can be used as a process example for this problem. In this case, as shown in FIG. 1, IND1 is the reboiler steam input, IND2 is the reflux rate, DEP1 is the C5+(pentane and heavier) concentration in the overhead product stream, and DEP2 is the C4−(butane and lighter) concentration in the bottoms product stream. In this example problem, the relative effects on the two product qualities are very similar, from a gain ratio perspective, regardless of which independent variable is manipulated. When reboiler steam is increased, the C5's in the overhead increase, and the C4's in the bottoms product decrease. When the reflux rate is increased, the C5's in the overhead product decrease, but the C4's in the bottoms product increase. The two independent variables have similar, but opposite, effects on the two dependent variables.

The gain matrix represents the interaction between both independent variables and both dependent variables.

	TABLE 1
	DEP1	DEP2
	(% C5+ Ovhd)	(% C4− Btms)
	IND1	37	−27
	(Reboiler Steam)
	IND2	−30	22
	(Reflux Rate)

The formula for Relative Gain Array is:

RGA(G)=G×(G⁻¹)^T   (1)

If the RGA formula is applied to our example 2×2 problem, the result is the 2×2 array:

TABLE 2
203.5  −202.5
−202.5 203.5

These RGA elements have a very high magnitude, which is undesirable. If the maximum acceptable RGA element magnitude is chosen to be 18, for example, the following formula can be used to calculate the logarithm base that will be used to modify the matrix.

$\begin{array}{cc}\mathrm{LOGBASE}=\frac{1}{\left[1-\frac{1}{\mathrm{MAX\_RGA}}\right]}=\frac{1}{\left[1-\frac{1}{18}\right]}=1.0588235& \left(2\right)\end{array}$

For each gain in the original matrix, the logarithm of the absolute value of the number with the base chosen from above (1.0588235 . . . ) is calculated, resulting in the matrix given in Table 3.

	TABLE 3
	DEP1	DEP2
IND1	63.17386488	57.66144728
IND2	59.50475447	54.07852048

In the preferred embodiment, each of these numbers is rounded to the nearest integer. The formula provided in equation 2 applies to the case where the rounding desired is to the nearest whole number (integer). In the event that rounding is desired to the nearest single decimal ( 1/10), then multiply the LOGBASE calculated in equation 2 by 10. In the event that rounding is desired to the nearest two decimals ( 1/100), then multiply the LOGBASE calculated in equation 2 by 100. This method is applicable to any degree of decimal precision by simply mutiplying the LOGBASE calculated in equation 2 by the 10 raised to the power corresponding to the number of decimals desired. The resulting integer matrix is shown in Table 4.

	TABLE 4
	DEP1	DEP2
IND1	63	58
IND2	60	54

The gains are recalculated by taking the logarithm base from formula (2) to the integer powers shown in TABLE 4. Where the original gain was a negative number, the result is multiplied by −1. Applying these steps results in the modified gain matrix shown in Table 5.

	TABLE 5
	DEP1	DEP2
IND1	36.63412093	−27.52756876
IND2	−30.86135736	21.90148291

If the RGA formula is applied to this matrix, the highest RGA element magnitude is equal to our desired maximum value shown in Table 6.

TABLE 6
−17 18
18  −17

The matrix modification process was able to do this by making relatively small changes in the original gain matrix. On a relative basis, the amount of gain change in each of the individual responses is shown in Table 7 below. This amount of change is normally well within the range of model accuracy.

	TABLE 7
	DEP1	DEP2
IND1	−0.99%	1.95%
IND2	2.87%	−0.45%

In an alternative embodiment, the base logarithm number can be chosen based on the maximum desired gain change, in units of percentage, using the formula (3) below. For the example problem used above, a maximum gain change of approximately 2.9% results in the same logarithm base as chosen above.

$\begin{array}{cc}\mathrm{LOGBASE}={\left[\frac{\mathrm{MAX\_CHNG}}{100}+1\right]}^2& \left(3\right)\end{array}$

In another alternative embodiment, the logged gains can be rounded to any fixed number of decimals for all matrix elements being operated on. For ease of use, it makes sense to choose a base logarithm where the desired results can be obtained from rounding the logged gains to an integer value. However equivalent results are obtained by rounding to any number of decimals if the base logarithm is adjusted. For example, if the base logarithm in the above example is chosen to be a power of ten greater than before,

LOGBASE=1.0588235¹⁰=1.77107   (4)

an equivalent result will come from rounding the logarithms of the gains to the nearest tenth.

In another alternative embodiment, the rounded numbers can be chosen to enforce a desired collinearity condition. If the difference between the rounded logarithms of the gains for two independent variables is the same for two different dependent variables, then that 2×2 sub-matrix is collinear. In other words, it is has a rank of one instead of two. The direction of rounding can be chosen to either enforce collinearity, or enforce non-collinearity. If the direction of rounding the logarithms of the gains from Table 3 is chosen to enforce collinearity, the integers could be chosen as shown in Table 8.

	TABLE 8
	DEP1	DEP2
IND1	63	58
IND2	59	54

The resulting matrix obtained by recalculating the gains is of rank 1 as shown in Table 9.

	TABLE 9
	DEP1	DEP2
IND1	36.63412093	−27.52756876
IND2	−30.86135736	21.90148291

Included in the preferred embodiment is the application of the same algorithm to any gain multiplication factor used inside the predictive model. Often gain multiplication factors are used to modify the model in response to changing conditions. Choosing the gain multiplication factor to be a rounded power of the same base as the model, will guarantee that the gain multiplied model has the same overall RGA characteristics.

Included in the preferred embodiment is the application of the same algorithm to building block models that are used to construct the final predictive model. Often the final model is the result of some combination of building block models that do not exist in the final application. By applying this same process to these building block models, the final model will have the same RGA characteristics.

The above description and drawings are only illustrative of preferred embodiments of the present inventions, and are not intended to limit the present inventions thereto. Any subject matter or modification thereof which comes within the spirit and scope of the following claims is to be considered part of the present inventions.

Claims

1. A method to modify a model gain matrix having at least one independent-dependent variable pair comprising: (a) choosing a logarithm base,(b) reading model gains for each independent-dependent variable pair,(c) taking the logarithm of the absolute value of the gain for each independent-dependent variable pair with the logarithm base chosen in step (a),(d) rounding the number from step (c) to a fixed number of decimals,(e) recalculating the gains by taking the logarithm base of step (a) raised to the power of the specified round-off from step (d),(f) multiplying the result of step (e) by −1 if the model gain was originally a negative number,(g) Applying these calculated gains to the model gain matrix.

2. The method of claim 1 wherein step (a) is performed by choosing a maximum allowable Relative Gain Array element (MAX_RGA) determined by the following formula.

3. The method of claim 1 wherein step (a) is performed by choosing a maximum allowable percentage gain change (MAX_CHNG) determined by the following formula.

4. The method of claim 1 wherein the direction of rounding performed in step (d) is chosen to force collinearity in 2×2 sub-matrices made up from two independent variable—dependent variable pairs.

5. The method of claim 1 wherein the direction of rounding performed in step (d) is chosen to force non-collinearity in 2×2 sub-matrices made up from two independent variable—dependent variable pairs.

6. The method of claim 1 further applied to internal gain multiplication factors used to modify the model gains.

7. The method of claim 1 further applied to building block models, which do not exist in the final model but are used to construct the final model.

8. The method of claim 1 in which the model is used for a multivariable predictive control application.

9. The method of claim 8 where the multivariable predictive control application is selected from the group of DMCplus and RMPCT.

10. The method of claim 1 in which the model matrix is used as an input to a linear program.

11. The method of claim 1 applied as a pre-processing step to multivariable predictive control calculations.

12. The method of claim 1 applied as a pre-processing step to planning and scheduling calculations.

13. The method of claim 1 where the rounding in step (d) is to zero decimals.

14. The method of claim 8 where the multivariable predictive control application is applied to control a manufacturing process.

15. The method of claim 14, where the manufacturing process is at least one petroleum refinery process selected from the group of refinery distillation unit, chemical plant distillation unit, crude distillation unit, vacuum distillation unit, naphtha reformer, naphtha hydrotreater, gasoline hydrotreater, kerosene hydrotreater, diesel hydrotreater, gas oil hydrotreater, hydrocracker, delayed coker, Fluid Coker, Flexicoker, steam reformer, sulfur plant, sour water stripper, boiler, water treatment plant and combinations of the above.

16. The method of claim 2 where the rounding in step (d) is to one decimal and where LOGBASE is multiplied by 10.

17. The method of claim 2 where the rounding in step (d) is to two decimals and where LOGBASE is multiplied by 100.

18. The method of claim 3 where the rounding in step (d) is to one decimal and where LOGBASE is multiplied by 10.

19. The method of claim 3 where the rounding in step (d) is to two decimals and where LOGBASE is multiplied by 100.