barcode reader integration with asp.net For m(T). To compute m(T), we change n in Fq (27.24) to 1 to obtain in Software

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For m(T). To compute m(T), we change n in Fq (27.24) to 1 to obtain
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m(T) = se(T) - aye(O) - bm(0) + (1 + b)m(-T) (27.26)
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Substituting the appropriate values of e and m as given in the table into this expression gives m(T) = 0.
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For m(2T). Letting n = 2 in Eq. (27.24) gives
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m(2T) = ae(2T) - aye(T) - bm(T) + (1 + b)m(O) (27.27)
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At t = 2T, the disturbance has worked its way through the transport lag and the error now differs from zero. We have at t = 2T as shown in the table or in Fig. 27.4 e(2T) = e(T) = m(T) = m(O) = -(l - b) 0 0 0
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Substituting these values into Q. (27.27) gives m(2T) = a[ -(l - b)]. Introducing the expression for (Y from EQ. (27.22) gives m(2T) = -(l + b + b*)
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For m(3T). Letting n = 3 in Eq. (27.24) gives
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m(3T) = cye(3T) - qe(2T) - bm(2T) + (1 + b)m(T) At t = 3T, e(3T) e(2T) m(2T) m(T) = = = = -(l - b*) -(l - b) -(l + b + b*) 0 (27.28)
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Substituting these values into IQ. (27.28) gives m(3T) = CY[-(1 - b*)] - cwy[-(1 - b)] - b[-(1 + b + b*)]
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Reducing this expression algebraically gives m(3T) = - 1,
DESIGN OF SAMPLED-DATA CONTROLLERS
J--LO 0 12 3 4 5tlT-
FIGURE 27-5
Response under fast sampling, load, minimal pmtotype algorithm to a unit-step change in load (a = OS),
If one continues in this sequential manner, which is how the computer handles the computation, one can show that
m(4T) = m(5T) = m(G) = -** = -1
In other words, the manipulated variable reaches -1 at t = 3T, and remains at this value thereafter. A graph showing the response and the manipulated variable is shown in Fig. 21.5 for the case where a = 0.5. For this case b = e-* r = ,-o.5 = 0.606 and
m(2T) = -1.974
The manipulated variable m(nT) that results for this case is not surprising when one considers the nature of the first-order system with transport lag. In fact, for this simple process, one can calculate the values of manipulated variable directly, without use of Eq. (27.24). However, for other disturbances and for more complex algorithms, the calculation becomes very involved without a systematic approach such as that given by Eq. (27.24).
Settling 7lme
A useful parameter for describing a transient response of a control system is settling time. For a load change, the settling time is defined as the time required to reduce the error to zero; this time is measured from the sampling instant for which nonzero error is recorded to the sampling instant for which the output returns to the set point and remains at the set point at future sampling instants. For the example under consideration, the settling time ts is 2T. This can be seen most easily from Fig. 27.5.
4 1 6 SAMPLED-DATAcONTROLSYSTEMS
For a set-point disturbance, the settling time is defined as the time required to reduce the error to zero; this time is measured from the sampling instant for which the set-point change is first detected to the sampling instant for which the response returns to the set point and remains there at future sampling instants.
OBTAINING M(z) DIRECTLY FROM KNOWLEDGE OF C(z). If one wishes to
compute M(z) without using the sequential method just discussed and shown in Table 27.1, the following direct procedure can be used. For the servo problem, where U(S) = 0, one can write directly from Fig. 27.3
C(z) = M(z)G(z)
(27.29)
where G(z) = GhGp(z) Solving for M(z) gives M(z) = C(z)/G(z) (27.30)
For the regulator problem, where R(s) = 0, one can obtain directly from Fig. 27.3 M(z) = -D(z)C(z) (27.31)
For Example 27.1, one can use Eq. (27.31) to obtain M(z) for a unit-step change in load; C in Eq. (27.3 1) is replaced by Cd of Eq. (27.18). The reader should try this approach to see that it leads to the same results as those obtained in Table 27.1.
Example 27.2. (T > UT, slow sampling, load)
In this example, the minimal prototype D(z) will be designed for the following conditions: T > a7 (slow sampling) U = l/s (load disturbance) G, = eCars/(~s + I), ur < T The fact that UT is not an integral number of sampling periods will make the derivation of this algorithm more complicated. Based on the response of a first-order system with transport lag, the minimal prototype response shown in Fig. 27.6 can be written:
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