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LINEAR CLOSED-LOOP SYSTEMS
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would cause T to run away and control would not be achieved. For this reason, positive feedback would never be used intentionally in the system of Fig. 9.2. However, in more complex systems it may arise naturally. An example of this is discussed in Chap. 21.
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Servo Problem versus Regulator Problem
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The control system of Fig. 9.2 can be considered from the point of view of its ability to handle either of two types of situations. In the first situation, which is called the servomechanism-type (or servo) problem, we assume that there is no change in load Ti and that we are interested in changing the bath temperature according to some prescribed function of time. For this problem, the set point TR would be changed in accordance with the desired variation in bath temperature. If the variation is sufficiently slow, the bath temperature may be expected to follow the variation in TR very closely. There are occasions when a control system in the chemical industry will be operated in this manner. For example, one may be interested in varying the temperature of a reactor according to a prescribed timetemperature pattern. However, the majority of problems that may be described as the servo type come from fields other than the chemical industry. The tracking of missiles and aircraft and the automatic machining of intricate parts from a master pattern are well-known examples of the servo-type problem. The other situation will be referred to as the regulator problem. In this case, the desired value TR is to remain fixed and the purpose of the control system is to maintain the controlled variable at TR in spite of changes in load Ti. This problem is very common in the chemical industry, and a complicated industrial process will often have many selfcontained control systems, each of which maintains a particular process variable at a desired value. These control systems are of the regulator type. In considering control systems in the following chapters, we shall frequently discuss the response of a linear control system to a change in set point (servo problem) separately from the response to a change in load (regulator problem). However, it should be realized that this is done only for convenience. The basic approach to obtaining the response of either type is essentially the same, and the two responses may be superimposed to obtain the response to any linear combination of set-point and load changes.
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DEVELOPMENT OF BLOCK DIAGRAM
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Each block in Fig. 9.2 represents the functional relationship existing between the input and output of a particular component. In the previous chapters, such inputoutput relations were developed in the form of transfer functions. In block-diagram representations of control systems, the variables selected are deviation variables, and inside each block is placed the transfer function relating the input-output pair of variables. Finally, the blocks are combined to give the overall block diagram. This is the procedure to be followed in developing Fig. 9.2.
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Consider first the block for the process. This block will be seen to differ somewhat from those presented in previous chapters in that two input variables are present; however, the procedure for developing the transfer function remains the same. An unsteady-state energy balance* around the tank gives
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q + WC(Ti - To) - WC(T - To) = PCVg (9.1)
where To is the reference temperature. At steady state, dT/df is zero, and Eq. (9.1) can be written
9s + wC(Ti, - To) - wC(T~ - To) = 0 (9.2)
where the subscript s has been used to indicate steady state. Subtracting Eq. (9.2) from Eq. (9.1) gives
4 - 9s + wC[(Ti - Ti,) - (T - Ts)l = PCV
d(T - Ts)
(9.3)
Notice that the reference temperature T, cancels in the subtraction. If we introduce the deviation variables
T/ = Ti - Ti, (9.4) (9.5) (9.6)
Q=s-4s
T = T - T ,
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