barcode reader integration with asp.net c(3T) - c(a) = [K(l - b) - b12 c(T) - cP9 in Software

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c(3T) - c(a) = [K(l - b) - b12 c(T) - cP9
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It is interesting to note that the relationship between decay ratio and fractional overshoot for the first-order, underdamped sampled system is the same as that for the second-order, underdamped continuous system, namely: Decay ratio = (fractional overshoot)2 (26.31)
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For a decay ratio (Ye, where 0 < (Y < 1, the loop gain may be computed from Eq. (26.29) to be
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c(r)
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for K= 1.72 (underdamped)
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1) forK=
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0.2 (overdamped)
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T=l z =I.25 T/z= 0.8
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FIGURE 26-5
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Transient response of a first-order, sampleddata system (no transport lag).
SAMPLED-DATA CONTROL OF A FIRST-ORDER PROCESS WITH TRANSPORT LAG
For any choice of sampling rate, one may determine from Eq.(26.32) the loop gain required for a desired decay ratio. Increasing the sampling rate (lower T) increases the speed of the response and decreases the period, which is 2T. This increase in sampling rate will also provide a larger open-loop gain, which gives less steady-state offset. From Eqs. (23.37) and (26.32), we may write K,2 - a+b_ (26.33) l+b KU For quarter-decay ratio ((Y* = i), one obtains from Eq. (26.33) for T ranging from 0 to 03
;<y<; u
(26.34)
TheratiovariesfromiforT =mto!forT =O. One may contrast this result with the Ziegler-Nichols rule for proportional control of continuous systems (see Chap. 17). The Ziegler-Nichols rule requires that K/K, = 4; if this rule is used for a continuous system, one expects to obtain a good transient response, for which quarter-decay ratio is often considered good in industrial practice. We see from Eq. (26.34) that K ,,djKu varies from 1 to i as the sampling period varies from infinity to zero. The Ziegler-Nichols rule and the rule provided by Eq. (26.34) are comparable.
Example 26.1. A simple example will help illustrate the use of the design equation,
Eq. (26.32). For the sampled-data system shown in Fig. 26.6, determine the openloop gain K for quarter-decay ratio for the following sampling periods: (a) T = 1, (b) T = 0.5, and (c) continuous control. Also find the period of oscillation and the offset for each case. b = e-T ~ = ,-0.8 = 0.449 (a) T = 1.25 T = 1.0 For quarter-decay ratio, a2 = 0.25 or (Y = 0.5 Substituting LY into Eq. (26.32) gives 1 - 0.449 From Eq. (26.28), the ultimate value of c is K c(c0) = - = 1.724 = 0.633 _ _
K+l 2.724 K
= a+b = 0.5 + 0.449 = 1 . 724 l-b
offset = r(m) - c(m) = 1 - 0.633 = 0.334 period = 2T = 2
FIGURE 26-6
Sampled-data control system for Example 26.1.
SAMPLED-DATA CO NT R O L S Y S T E M S
(b) For this case, where T/r = 0.4 and b = 0.670, the answers are: K ,, j = 3.550 offset = 0.219 period = 2T = 1 (c) For continuous proportional control of a first-order process, the transient response is never oscillatory; therefore, ELq. (26.32) does not apply. This example illustrates that a faster sampling rate permits a higher proportional gain and less offset for a fixed decay ratio.
Case II: lkansport Lag, 0 < UT I T, n = 0
For this case, the response will be delayed by an amount UT . To determine the decay ratio and overshoot, we must be able to compute the peaks of the transient response. Because of the delay ar, the peaks will not occur until UT after the sampling instants. This observation is based on an understanding of the behavior of a first-order system and the transport lag. The sketch shown in Fig. 26.7 illustrates the situation. To determine the peak values, we must invert C(z,m) with AT = T - a7 and m = 1 - A = ar/T As shown in Eq.(26.23), we must obtain G(z,m) in order to determine C(z,m). The transfer function G(s) is - e-TS) (26.35) s We cannot obtain G(z,m) for G(s) in Eq. (26.35) directly from the table of transforms because G(s) contains eeaTs where ~7 is a nonintegral number of sampling periods (i.e., UT < T). However, we can obtain G(z,m) by using the approach taken in developing the modified Z-transform in Chap. 25. We express G(s) as GA(S) where GA(S) = G(s)e --h7S. Recall that AT is the amount by which the response is to be shifted. We see from Fig. 26.7 that AT = T - a~. Now GA(S) can be written G(s) = GcG,fQs) = s
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