barcode reader project in c#.net Overall lhnsfer hnction for Single-Loop Systems in Software

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Overall lhnsfer hnction for Single-Loop Systems
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Once a control system has been described by a block diagram, such as the one shown in Fig. 12.1, the next step is to determine the transfer function relating C to R or C to U. We shall refer to these transfer functions as overall transfer functions because they apply to the entire system. These overall transfer functions are used to obtain considerable information about the control system, as will be demonstrated in the succeeding chapters. For the present it is sufficient to note that they are useful in determining the response of C to any change in R and U.
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FIGURE 12-2
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Block diagram for a multiloop, multiload system.
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FIGURE 12-3 Block-diagram reduction to obtain overall transfer
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The response to a change in set point R, obtained by setting U = 0, represents the solution to the servo problem. The response to a change in load variable U, obtained by setting R = 0, is the solution to the regulator problem. A systematic approach for obtaining the overall transfer function for set-point change and load change will now be presented.
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Overall Ikansfer Function in Set Ebint
for Change
For this case, U = 0 and Fig. 12.1 may be simplified or reduced as shown in Fig. 12.3. In this reduction, we have made use of a simple rule of block-diagram reduction which states that a block diagram consisting of several transfer functions in series can be simplified to a single block containing a transfer function that is the product of the individual transfer functions. This rule can be proved by considering two noninteracting blocks in series as shown in Fig. 12.4. This block diagram is equivalent to the equations Y -= GA X Multiplying these equations gives YZ - - = GAGB XY which simplifies to Z - = GAGB X Thus, the intermediate variable Y has been eliminated, and we have shown the overall transfer function Z/X to be the product of the transfer functions GAGS. This proof for two blocks can be easily extended to any number of blocks to give the rule for the general case. This rule was developed in Chap. 7 for the specific case of several nonintemcting, first-order systems in series. Z - = GB Y
x-@++z
FIGURE 12-4 ltvo aoaiateractiag blocks ia series.
LINEAR
cLosED-L.ooP SYSTEMS
With this simplification the following equations can be written directly from Fig. 12.3b. C = GE B = HC E=R-B (12.1) (12.2) (12.3)
Since there are four variables and three equations, we can solve the equations simultaneously for C in terms of R as follows: C = G(R - B) C = G(R-HC) C = GR-GHC or finally
C G -=-
l+GH
(12.4)
This is the overall transfer function relating C to R and may be represented by an equivalent block diagram as shown in Fig. 12.3~.
Overall Transfer Fbnction for Change in Load
In this case R = 0, and Fig. 12.1 is drawn as shown in Fig. 12.52. From the diagram we can write the following equations: C = M= B = E = G2(U+M) G,Gle HC -B (12.5) (12.6) (12.7) (12.8)
Again the number of variables (C, U, M, B, E) exceeds by one the number of equations, and we can solve for C in terms of U as follows: C = G2(U + G,Gle) C = G2[U + G,GI(-HC)]
FIGURE 12-5 Block diagram for change in load.
CLOSED-LOOP
TRANSFER FUNCI IONS
or finally (12.9) where G = GcGiG2. Notice that the transfer functions for load change or setpoint change have denominators that are identical, 1 + GH. The following simple rule serves to generalize these results for the singleloop feedback system shown in Fig. 12.1: the transfer function relating any pair of variables X, Y is obtained by the relationship Y Tf -=negative feedback X 1 + lr[ where rrf = product of transfer functions in the path between the locations of the signals X and Y 7~1 = product of all transfer functions in the loop (i.e., in Fig. 12.1, al = G,G1G2H) If this rule is applied to finding C/R in Fig. 12.1, we obtain C G GcGG -= R .l + G,GIG2H = l+GH which is the same as before. For positive feedback, the reader should show that the following result is obtained: Y f -=positive feedback (12.11) X 1 - TT[
Example 12.1. Determine the nansfer functions C/R, C/Ul, and B/U2 for the system show in Fig. 12.6. Also determine an expression for C in terms of R and Ur for the situation when both set-point change and load change occur simultaneously. Using the rule given by Eq. (12.10), we obtain by inspection the results
(12.12)
(12.13) (12.14)
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