1.8 Block Diagram Reduction The discussion of Section 1.7 appears to imply that if the transfer function relating input r and output c in block diagram， such as Fig .1.1 is desired， a differential equation relating these two variables must... 1.8 Block Diagram Reduction
The discussion of Section 1.7 appears to imply that if the transfer function relating input r and output c in block diagram， such as Fig .1.1 is desired， a differential equation relating these two variables must be obtained first. Fortunately ， this is not necessary. The transfer function can be derived instead by certain algebraic manipulations of those of the subsystems or blocks. Some examples will show this block diagram reduction technique and provide some useful results.
Example 1.8.3 The configuration in Fig.1.5(a)，which includes a minor feedback loop， is very common in servomechanisms . Derivation of C/R by the approach of Example 1.8.2 would be laborious ，but become simple if the result in(1.33) is used. It is applied first to reduce the minor feedback loop C/M to a single block ， as shown in Fig.1.5(b). but (1.33) applies again to this new loop and now yields the closed-loop transfer function.
Example 1.8.4 In a tow-input system， the additional input D often represents a dis-turbance ， such as a supply pressure variation in the level control example in Section 1.3 . With the additional block L ， the diagram models the effect of the disturbance on the system. For linear systems the principle of superposition applies， and the total output is the sum of the outputs due to each input separately. Thus the out-put due to R is found as before， and while finding that due to D， R is put equal to zero. The rule of Example 1.8.2 applies when finding the response to D， but note that the product of G2. Note also that for R=0 the minus sign for the feedback at R can be moved to the summing junction for D. Inspection now yields.
Example 1.8.5 In fig.1.6 the two feedback loops interfere with each other. The rearrangements (a) and (b) are alternative first steps to make the result in (1.33) again applicable . Verify that neither changes the system， and that applying (1.33) twice to (a) or (b) yields the closed-loop transfer function.
1.9 Conclusion
In this chapter a general introduction has been given first， including physical discussion of some fundamental features of control system behavior. A level control example led to a common block diagram configuration. Laplace transforms led to the transfer function description of dynamic behavior， and block diagram reduction to the description of an interconnected system of blocks. The application of transfer functions and transforms and transforms to calculation of the response c(t) to an input r(t) and initial conditions has been demonstrated for cases where the roots of the denominator of the transform C(s) are real and distinct. This provides a framework and motivation for study of the next chapter， and a basis for detailed discussion of transient response in Chapter 3. It also allows for an introductory examination of some of the effects of feedback in the problems below. 展開
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