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FIG 8.7. Lack of dynamic compensation produces a transient equal to the difference in dead times.
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If only a steady-state control calculation is made, 63.6) Differentiating,
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= k (r dq + q dr) P
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(8.7) in Eq. (8.5) yields the closed-loop response: (8.8)
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dc (t) = dr (t - T,,J + dq ; (T, - em)
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Equation (8.8) shows that the set-point response is delayed by TV and that a load change will induce a transient of duration TV - 7n and magnitude r dq/q. Both responses appear in Fig. 8.7. Of the two, load response is the more important, because set-point changes are ordinarily less frequent. Ideally, the load signal should be delayed by 7q before it is multiplied, and then advanced by rm. It is impossible to create a time advance, however. So dynamic compensation is best introduced in this application by delaying the feedforward signal by an amount 7q - 7n. If 7m > 7q, compensation is impossible. It has been pointed out that dynamic compensation generally takes the It may be recalled, however, that the ratio of two vector form g,/g,. quantities like these resolves into the ratio of their magnitudes and the difference between their phase angles. Since dead-time elements have unity gain, their ratio is also unity; their only contribution is phase lag. This is why the ratio g,/g, appears as the difference 79 - , between the dead times. The complete forward loop, including dynamic compensation, appears in Fig. 8.8. Note the complete cancelation of all elements in the load path by the elements in the forward loop.
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214 1 Multiple-loop Systems
Observe how faithfully the forward loop images the properties
Because forward loops exhibit absolutely no oscihatory tendencies, to talk of gain and phase is rather inconsequential. Step responses will be used throughout, since they constitute the most severe test of system performance. The response of systems under feedforward control, both with and without dynamic compensation, differs markedly from that experienced with feedback control. For this reason, it is not surprising that dynamic elements in the forward loop bear little resemblance to the conventional modes of feedback controllers. Although dead time serves as a useful demonstration of why dynamic compensation is necessary, it rarely appears alone in a process. In fact, multiple lags are most, commonly encountered in actual applications. Fortunately, there is usually one dominant lag on each side of the process, which acts as the principal element to be compensated. The response of a process wherein g, and g, are first-order lags of time constants TV and r4, respectively, can be found by substituting their individual response terms into Eq. (8.8). Thus t - 7m becomes 1 - e+ m, and 79 - 7m is replaced with e--t/~, _ e-t/~,,,: & (t) = dr (1 - e-t/rm) + dq i (e-t/r, - e-t/Tm) (8.9)
Figure 8.9 gives both set-point and load-response curves described by this equation, for the case where 7q > r,,,. Compare it to the heat-exchanger response, Fig. 8.5, where rm > rq.
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FIG 8.9. Lack of dynamic compensation shows up principally as a load-response transient.
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FIG 8.10. Comparison of the openloop response of c to m with the reverse response of c to q shows that m must be made to lag q for this process.
3 4 Time, min
A qualitative appraisal of the requirement for dynamic compensation may be obtained from a comparison of open-loop response curves. Because an increase in the manipulated variable acts in opposition to the load, their individual step-response curves will diverge. One or the other response will have to be inverted so that the two curves may be superimposed, as is done in Fig. 8.10. The response of such a process under uncompensated feedforward control appears as the difference between these two curves. If the curves do not cross, the uncompensated forward-loop response will lie wholly on one side of the set point, as in Figs. 8.5 and 8.9. Which side of the set point depends on whether the difference g, - g, is positive or negative. If the curves cross, the uncompensated forward-loop transient will cross the set point.
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