Multivariable Process Control in .NET framework

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Multivariable Process Control
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FIG 7.5. Half-coupling exists between composition x and flow Y.
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Half-coupling
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It is possible to arrange a 2 by 2 process such that one variable will be affected by one valve while the other is influenced by both. If the flow measurement for the blending system were placed on stream Y as in Fig. 7.5, this effect would be achieved. Both valves now affect composition, but ~21 has no influence over flow Y: Y=m2 g=Q
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(7.17) The remaining ele-
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The first element in the matrix, XyI, must be zero. ments are: nzl lrl2
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The matrix indicates that coupling is not a problem in this process. And indeed this is the case. First, there is no question about the pairing of variables, as there is with fully coupled loops. Second, although UL~ Finally, influences composition, the effect is more that of a load change. since the composition controller cannot cause 111~ to change by altering the input to the flow controller, its adjustment depends only on the gain through 7~2~. Notice that the blending system in Fig. 7.1 is half-coupled.
Dynamic Effects
In the pressure-flow process, the dynamic response of both measurements is similar. This is not often the case. Flow, pressure, and liquid level ordinarily respond rapidly to valve position, while temperature and composition do not. Therefore processes which couple flow and composition, level and composition, pressure and t.emperature, etc., are deserving of deeper study. Actually, the coupling of a fast and a slow loop was simulated in the earlier example when the pressure controller was given loose settings while the flow controller was tuned tightly. Imagine the settings of the pressure controller being more like those of a composition controller.
1 9 6 1 Multiple-loop Systems
The fast loop is scarceIy affected by the coupling, while the slower loop is upset by the fast one. The blending system of Fig. 7.4 is a good example of the coupling of fast and slow loops. The composition loop would normally oscillate at a period of a few minutes, while the period of the flow loop would be a few seconds. Tuning of the flow controller would be the same as if the loops were entirely independent, because the composition controller would not be able to cause rapid changes in flow. But any increase in ml will increase total flow, thereby automatically decreasing m2. The composition controller actually manipulates both valves-one directly, t he other indirectly through the flow controller. Consequently it must be adjusted to accommodate the gain of both valves. An appreciation for the dynamic effects that coupled closed loops have on one another can be gained by analyzing the block diagram shown in Fig. 7.6. One loop, comprised of a cont.roller whose gain vector is g,, and a process with a dynamic vector of g, and relative gain XII, has a period of ~~1. The loop is upset by a manipulated variable 112~ from the second closed loop, whose period is 70z. In the path of 1722 is the process vector g2 and the relative gain of c1 with respect to 7n2, that is, X12. Although the solution of the block diagram for both closed loops is unwieldy and difficult to present, a qualitative appraisal of the dynamic effect can be gained from the response of cl with respect to m2.
fnl = gcl(rl Cl(l + - Cl) = Cl = mg1X11 + + 77zzg2~12
(7.19)
gclglX11)
~lgclgixll
77~2gz~12
Differentiation of cl with respect to dc1 dm2
Ocr 4 -= 1 + gzx12 gelgAl1
yields (7.20)
FIG 7.6. Each closed loop is upset by the output of the other.
Multivariable Process Control
FIG 7.7. The effect of m2 on cl depends on the ratio of their periods.
Equation (7.20) can be solved by factoring into two parts, g,xlz and l/(1 + g,lglX1l). The latter is a characteristic of the closed loop and can be evaluated by summing the open-loop vector g,,g,X1l with 1.0, followed by inversion. The solution for the general case of a dead-time plus integrating combination adjusted to g,,g, = 0.5 at the natural period is plotted in Fig. 7.7. When 702 is less than 0.5701, dynamic coupling is virtually unity. This means that any change in the output of controller 1 will affect both ml and m2, such that its gain must be reduced by an amount corresponding to x12. When there are only two loops, the final gain becomes X1* times the original. Values of r02 between 0.5 and 2r0r represent, a region of severe dynamic coupling. If 7,2 falls in this area, the gain of controller 1 must be reduced by at least X12, which reduces the magnitude of the coupling. Figure 7.7 shows two curves, one where Xl1 is 1.0, i.e., the controller is adjusted for conditions of no coupling. A second curve is plotted for X11 = 0.5, which corresponds to a reduction in controller gain by 50 percent, allowing for the maximum coupling in a two-loop system. This makes the loop less sensitive to disturbances occurring in the resonant region, but more sensitive to disturbances at longer periods. Where ro2 >> rol, coupling approaches zero. This is the case of the flow loop being scarcely disturbed by the composition loop. In this region higher controller gain is actually favorable, reducing the effect of dynamic coupling.
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