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function of a system containing transport lag (e - ) for which T is not an integral number of sampling periods. It should be remembered that a modified Z-transform is simply a Z-transform of a function in which a transport lag (e -hTs) has been included. The inversion of a modified Z-transform is obtained in the same manner as the inversion of an ordinary Z-transform, by long division or by use of partial fraction expansion.
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25.1. For the control system shown in Problem P23.4, determine the response between
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sampling for the case m = 0.4 by use of the modified Z-transform.
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25.2. For the process shown in Fig. P25.2, determine Y(z). By means of long division,
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determine Y(t) for t = 0, 1, 2, 3, 4, and 5.
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25.3. (a) For the control system shown in Fig. P25.3, obtain a general expression for
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C(z) for the case where R = 0 (no set-point change) in terms of G 1, G2, and U. (b) One can show that for R = 0 and U = l/s that C(z,m = 0.3) = 0.2592 - + 0.587~-~ + 0.556~-~ + . . . . From this result and any other information you wish to use, determine if possible the values of C at the following times: 0,0.3,0.5,0.7, 1 .O, 1.3, 1.5, 1.7, 2.0, 2.3, and 3.0. Present your results in a table.
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FIGURE P25-3
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CHAPTER
SAMPLED-DATA CONTROL OF A FIRST-ORDER PROCESS WITH TRANSPORT LAG
The tools developed in the previous chapters for sampled-data systems will now be applied to a model found to fit a large class of systems in chemical processing. This model consists of a first-order process with transport lag (or delay). The transfer function may be written G,(s) = s (26.1)
where T is the time constant and UT is the transport lag parameter (a is a positive number). Consider the sampled-data control system shown in Fig. 26.1 in which this transfer function is used as a model of the process. In this discussion, the hold will be a zero-order hold for which the transfer function is 1 (26.2) G&) = $1 - e -Ts ) Recall that the combination of the sampling switch and the zero-order hold provides clamping. We shall take the control action to be proportional, for which
G, = K
(26.3) 393
SAMPLED-DATA
CONTROL
SYSTINS
Gc(s)
-an e 7s+l
l!lbUla - - - - - - L-6-1 Sampled-da ta control of a firstndl=r mfrtl=r -_--_ IJ ,,.n with transport lag.
For the case of no transport lag (a = 0), the proportional control of this process was discussed in Chap. 23. As will be shown, the higher the value of a, the higher the order of the characteristic equation for the closed-loop system. We shall consider a number of cases that are based on the size of a. For this purpose, let
nT < UT 5 (n + l)T
(26.4)
where n = 0, 1, 2, 3, . . . For convenience in obtaining G(z), ur may be written as follows:
= nT + (UT - nT)
(26.5)
In this form ur is equal to an integral number of sampling periods (nT) plus a fraction of a sampling period [i.e., (ur - nT) 5 T]. Using this expression for UT in Eq. (26. l), the open-loop transfer function becomes
e-(aT-nT)s
G(s) = K(l - e-r )Pr
S(TS
+ 1)
(26.6)
Using the theorem on translation in Chap. 23 [Eq. (23.26)], we may write G(z) as
G(z) = KU - z-9 z { $y+;;} Z"
(26.7)
Note that the expression within braces is equivalent to delaying the response from l/[S(TS + l)] by (U T - n T). We may apply the concepts used in developing the modified Z-transform to find the Z-transform of the expression within braces in Eq. (26.7) by equating ( UT - nT) to AT and recalling that m = 1 - A. This leads to
AT = UT - nT m = l - A = I--a$+n
(26.8)
Equation (26.7) may now be written
G(z) = KC1 - z-9
Z zm( S(TS1+ l)}
(26.9)
SAMPLED-DATA CONTROL OF A FIRST-ORDER
PROCFM W I T H T R A N S P O R T L A G
From the table of modified Z-transforms (Table 22. l), we obtain
G(z) = KU - z-9 - - exp[-$(l - y + n)T] 1
z - l gives = zn+l
z - e-T/r
(26.10) I
Rearranging
this
expression G(z)
K(z - 1)
and b = e-T 7 1 + G(z) = 0
where d = exp[a - (n + ~)T/T]
(26.11)
The characteristic equation for the closed-loop response may be written
or 1 + Kk. or
zn+l
1 d 1) - = 0 z - l z - b
(26.12)
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