barcode reader integration with asp.net where G(z) = G,Gh(z> We may also obtain from Fig. 27.3 in Software

Encoder Code 128 Code Set C in Software where G(z) = G,Gh(z> We may also obtain from Fig. 27.3

where G(z) = G,Gh(z> We may also obtain from Fig. 27.3
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UG,(z)
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E(z) = 1 + G(z)D(z) - 1 + G(z)D(z)
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(27.11)
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C(z) = R(z) - E(z) Combining Eqs. (27.11) and (27.12) gives
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(27.12)
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( ) = 1 + G(z)D(z)
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1 + G(z)D(z)
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(27.13)
SAMPLED-DATA CONTROL SYSTEMS
Note that R(z) in Eqs. (27.11) and (27.13) is not bound to other transfer functions. This can be advantageous in design computations.
Design Methods
It is convenient to design D(z) for a load change or a set-point change. For a given disturbance in load or set point, the designer proposes a desired response at sampling instants, which means that C(nT) must be specified. This desired response will be written as Cd. From the desired response Cd(nT), one obtains Cd(z). Consider the case of only a load disturbance, i.e., R(z) = 0. Solving Eq. (27.13) for D(z) and replacing C by Cd to indicate that the desired response of C has been selected by the designer give
D(z)= &[$$$l]
(Design equation for load change)
(27.14)
Equation (27.14) is the equation to be used to design D(z) for a load change. In a similar manner, one can obtain from Eq. (27.13) a design equation for set point change, i.e. U(z) = 0. The result is
Cd(Z) D(z) =
G(z)[R(z)
- cd(z)]
(Design equation for set-point change)
(27.15)
It is necessary that the highest power of z in the numerator of D(z) not exceed the highest power of z in the denominator. If this restriction is not satisfied, the algorithm will require knowledge of the future values of the error, i.e., prediction. An algorithm not satisfying this restriction is called unrealizable. Note that Eq. (27.4) is written in such a form that it does not admit the case where the highest power of z in the numerator exceeds the highest power of z in the denominator. If an unrealizable D(z) is obtained, we obtain the expression given by Eq. (27.4) multiplied by some positive power of z . To show how the control algorithms are obtained, several detailed examples that apply to Fig. 27.3 will be presented.
Example 27.1. (T = UT, fast sampling, load)
With regard to Fig. 27.3, consider the design of D(z) for T = a~. This relation between the sampling period and the process transport lag will be referred to as fast sampling. Later, an example involving slow sampling will be considered, in which T > aT. Consider a unit-step change in load to enter the system at an instant of sampling. The Z-transform of the output can be written in general form:
(27.16) c,(z) = 170 + qlz- + 72z-* + * where the coefficients (7/i) correspond to the desired outputs of the system at sampling instants. It is the task of the designer to specify a desired output that leads to a realizable algorithm and fulfills the performance specifications listed earlier. In some cases, the nature of the physical process being controlled will aid in choosing a suitable Cd(Z) as expressed by Eq. (27.16).
DESIGN
OF SAMPLED-DATA CONTROLLERS
The response diagram in Fig. 27.4 will help explain how the output C,(z) is selected by the designer. Because the transport lag in Gp(s) is one sampling period T, the output will remain at 0 for the first two sampling instants, that is, c(O) = 0, c(T) = 0 The response of c, which is the usual first-order exponential rise starting at t = T, will reach 1 - emTk at t = 2T, that is, (27.17) c(2T) = 1 - edT r = 1 - b where b = e- . The control algorithm in D(z) cannot start to respond until t = 2T, at which time the error e has become - (1 - b). The value of the manipulated variable m c generated by the algorithm at t = 2T will depend on the actual algorithm used. However, regardless of the value of m, at t = 2T, c will not change its course until t = 3T because of the transport lag emTS. The output of c from t = 2T to t = 3T will continue as an exponential rise and reach 1 - e-2T r at t = 3T, that is, c(3T) = 1 - e -2Th = 1 _ b2 After t = 3T, the designer is free to choose any values of c at sampling instants. If c is chosen as zero at sampling instants beyond t = 3T, the resulting algorithm will be called a minimal prototype algorithm because the output returns to the set point in a minimum number of sampling periods. The response shown in Fig. 27.4 illustrates the minimal prototype response for a unit-step change in load.
l- b2 LLL -------1-b - - - - -
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