Multivariable Process Control in Visual Studio .NET

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Multivariable Process Control
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corrected as the controller slowly moves valve ~1. This gradual movement of ~72~ will cause a slight flow change, causing t he flow controller to reposition ~11~. If the ratme of change of 112~ is slow compared to the reset, time of the flow controller, flow will be maintained quite near the set point throughout. Under these conditions, flow is slightly affected by pressure, while pressure is tightly coupled to flow. If -the pressure controller were now tightly adjusted with the flow controller in manual, both loops would be on the verge of instability when placed in automatic. An increase in set flow would cause 1n2 to open, dropping p,. The pressure controller would open lnl to restore pl, with the result that flow would increase twice as much as the flow controller intended it to. In effect, coupling has caused reverberation between the loops by doubling the gain of each. This factor should be evident from the matrix: the total effect on flow (and pressure) by both valves is 1.0, yet each manipulated valve only affects each measurement by 03. Thus the flow controller was originally adjusted with only half the process gain in force. If the matrix elements had been 0.2 and 0.8 instead of 0..5 and 0.5, the effect on the closed loops would have been hardly noticeable (unless the wrong pairs had been connected). The stability problem can be resolved by doubling the proportional band of each controller. Even so, a change in the set point of either controller will upset the ot her loop, because both valves must move. But if the elements were 0.8 and 0.2, and the wrong pairs of variables were chosen, the situation would be considerably worse. A flow set-point change would cause a fourfold upset in pressure, and vice versa. If the proper combinations were chosen, the upset would be He as great. T h i s example should point out how important it is to determine and select the proper pairing of variables for each loop. Another observation to be made is that the degree of coupling is usually variable. It varied with p in the pressure-flow process and with z in the blending system. In each case it depended on the extensive controlled variable rather than on the intensive variable (flow). It should be recognized that in some processes the best choice of C-W pairs at one operating condition may not be the best at another.
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Coupling between Similar Variables
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A further degree of coupling is found between variables that are similar in nature. Imagine a three-component, blend in which it is desired to control both density p and viscosity p. A problem arises because a change in either of two components may affect both density and viscosity in the Same direction. This differs from the cases studied earlier, in that while ml and m2 affected one variable in the same direction, they affected the other in opposite directions. Let the mathematical model of the three-
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194 1 Multiple-loop Systems
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component blending system be as follows:
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(7.14) (7.15)
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Let the coefficients a, b, f, and g be positive numbers, and total flow F be uncontrolled, so that it does not enter into the matrix. The normalization procedure yields the following matrix:
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ml m2
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ag - bf ag - bf ag - bf ag - bf
-bf ~
ag ~
Since all coefficients are positive, two of the terms in the relative-gain matrix must be negative-which two depending on whether ag > bf. To allow inspection of the properties of this process, let a = b = f = 0.5 and g = 1.0; then the matrix appears as:
-1 2
Pairing must still be carried out in favor of the larger (positive) numbers. In fact the pairing indicated by the negative numbers will not be controllable at all. If m2 were chosen for control of density and ml for viscosity, the manipulated variables would eventually be driven to opposite limits without satisfying either controller. Regardless of the controller settings, the system is divergent. This is always an indication of positive feedback. If the correct pairing is chosen, both loops will be stable and in fact will require double the controller gain needed if there were no coupling. Thus it can be seen that even in this case, the numbers in the matrix indicate the effect of coupling on controller settings. The coupling in this example is constant, i.e., only constants appear in the matrix, because the mathematical model of the process is linear. Observe how the coefficients in the model fall into place in the relativegain matrix, corresponding to the transformation procedure involving Eq. (7.13).
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