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FIGURE P13-4 (a) Draw a block diagram of this control system, and in each block give the transfer function, with numerical values of the parameters. (b) Determine the controller gain K, for a critically damped response. (c) If the tanks were connected so that they were interacting, what is the value of K, needed for critical damping (6) Using 1.5 times the value of Kc determined in part (c), determine the response of the level in tank 2 to a step change in set point of 1 in. of level. 13.5. A PD controller is used in a control system having a first-order process and a measurement lag as shown in Fig. P13.5. (4 Find expmzons for 6 and T for the closed-loop response. (b) If 71 = 1 min, TV = 10 set, find K, so that l = 0.7 for the two cases: (1) 7-0 = 0, (2) 70 = 3 sec. (4 Compare the offset and period realized for both cases, and comment on the advantage of adding the derivative mode.
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FIGURE P13-5 13.6. The thermal system shown in Fig. P13.6 is controlled by a PD controller. Data: w = 250 lb/min p = 62.5 lb/ft3
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&=4ft3 v2 = 5 ft3 V3 = 6 ft3
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LINEAR
CLQSED-LOOP
SYSTEMS
A change of 1 psi from the controller changes the flow rate of heat q by 500 Btuhnin. The temperature of the inlet stream may vary. The= is no lag in the measuring element. (a) Draw a block diagram of the control system with the appropriate transfer function in each block. Each transfer function should contain numerical values of the parameters. (b) From the block diagram, determine the overall transfer function relating the temperature in tank 3 to a change in set point. (c) Find the offset for a unit-step change in inlet temperature if the controller gain K, is 3 psi per OF of tempera- error and the derivative time is 0.5 min. 13.7. (a) For the control system shown in Fig. P13.7, obtain the closed-loop transfer function C/U. (b) Find the value of K, for which the closed-loop response has a f of 2.3. (c) Find the offset for a unit-step change in U if Kc = 4.
10.25s + 1 1 FIGURE P13-7 13.8. For the control system shown in Fig. P13.8, determine: (4 C(sYNs) (b) C(w) (c) offset (4 W.5) (e) whether the closed-loop response is oscillatory
FIGURE Pl3-8
s(s+l)
TRANSIENT
RESPONSE
SIMPLE
CONTROL
SYSTEMS
FIGURE P13-9
13.9. For the control system shown in Fig. P13.9, determine an expression for C(t) if a unit-step change occurs in R. Sketch the response C(f) and compute C(2). 13.10. Compare the responses to a unit-step change in set point for the system shown in Fig. P13.10 for both negative feedback and positive feedback. Do this for K, of 0.5 and 1.0. Compare these responses by sketching C(r).
FIGURE P13-10
CHAPTER
STABILITY
CONCEPT OF STABILITY
In the previous chapter, the overall response of the control system was no higher than second-order. For these systems, the step response must resemble those of Fig. 5.6 or of Fig. 8.2. Hence, the system is inherently stable. In this chapter we shall consider the problem of stability in a control system (Fig. 14.1) only slightly more complicated than any studied previously. This system might represent proportional control of two stirred-tank heaters with measuring lag. In this discussion, only set-point changes are to be considered. From the methods developed in Chap. 12 for determining the overall transfer function, we have from Fig. 14.1.
C KcG -= R 1 + K,GH
(14.1)
In terms of the particular transfer functions shown in Fig. 14.1, C/R becomes, after some rearrangement,
c -=
Kc(73S
(7,s + l)(~~s + l)(~~s + 1) + K,
+ 1)
(14.2)
The denominator of Eq. (14.2) is third-order. For a unit-step change in R, the transform of the response is
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