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where G = G,G 1 G2G3H 1 Hz. The reader should check one or mote of these results by the direct method of solution of simultaneous equations. For separate changes in R and Ut , we may obtain the response C from Eqs. (12.12) and (12.13); thus
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(12.15)
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If both R and Ut occur simultaneously, the principle of superposition requires that the overall response be the sum of the individual responses; thus
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(12.17)
Overall Ihnsfer Function for Multiloop Control Systems
To illustrate how one obtains the overall transfer function for a multiloop &em, consider the next example in which the method used is to reduce the block diagram to a single-loop diagram by application of the rules summarized by Eqs. (12.10) and (12.11).
Example 12.2. Determine the transfer function C/R for the system shown in Fig. 12.7. This block diagram represents a cascade control system, which will be discussed later.
RR Rjx/+C lb)
FIGURE 12-7
Block diagram reduction: (a) original diagram, (b) first reduction, (c) final single-block diagram.
CLOSED-LOOP TRANSFER FUNdONS
Obtaining the overall transfer function C/R for the system represented by Fig. 12.7~ is straightforward if we first reduce the inner loop (or minor loop) involving Gcz, G1, and H2 to a single block, as we have just done in the case of Fig. 12.1. For convenience, we may also combine G3 and G3 into a single block. These reductions are shown in Fig. 12.7b. Figure 12.7b is a single-loop block diagram that can be reduced to one block as shown in Fig. 12.7~. It should be clear without much detail mat to find any other transfer function such as C/U1 in Fig. 12.7a, we proceed in the same manner, i.e., first reduce the inner loop to a single-block equivalent.
SUMMARY
In this chapter, we have illustrated the procedure for reducing the block diagram of a control system to a single block that relates one input to one output variable.
This procedure consists of writing, directly from the block diagram, a sufficient number of linear algebraic equations and solving them simultaneously for the transfer function of the desired pair of variables. For single-loop control systems, a simple rule was developed for finding the transfer function between any desired pair of input-output variables. This rule is also useful in reducing a multiloop system to a single-loop system. It should be emphasized that regardless of the pair of variables selected, the denominator of the closed-loop transfer function will always contain the same term, 1 + G, where G is the open-loop transfer function of the single-loop control system. In the succeeding chapters, frequent use will be made of the material in this chapter to determine the overall response of control systems.
PROBLEMS
12.1. Determine the transfer function Y(s)/X(s) for the block diagrams shown in Fig. P12.1. Express the results in terms of Ga, Gb, and G,.
(b) FIGURE P12-1
LmEAR
cl.mFD-LOOP SYSTFMS
12.2. Find the transfer function Y(s)/X(s) of the system shown in Fig. P12.2.
FIGURE P12-2
12.3. For the control system shown in Fig. P12.3 determine the transfer function C(s)/R(s). .
FIGURE P12-3
12.4. Derive the transfer function Y/X for the control system shown in Fig. P12.4.
FIGURE P12-4
CHAPTER
TRANSIENT RESPONSE OFSIMPLE CONTROL SYSTEMS
In this chapter the results of all the previous chapters will be applied to determining the transient response of a simple control system to changes in set point and load.* Considerable use will be made of the results of Chaps. 5 through 8 (Part II) because the overall transfer functions for the examples presented here reduce to first- and second-order systems. Consider the control system for the heated, stirred tank that has been discussed in Chaps. 1 and 9 and is represented by Fig. 13.1. The reader may want to refer to Chap. 9 for a description of this control system. In Fig. 13. la, the sketch of the apparatus is drawn in such a way that the source of heat (electricity or steam) is not specified. To make this problem more realistic, we have shown in Fig. 13. lb that the source of heat is steam that is discharged directly into the water and in Fig. 13. lc the source of heat is electrical. In the latter drawing, a device known as a power controller provides electrical power to a resistance heater proportional to the signal from the controller.
*The reader who is interested in the simulation of control systems by digital computer is advised to study Chap. 34 at this point.
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