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Thermocouple installation for heated-tank system.
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INTRODUCTORY
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Tank temperatmt versus time with measuring lag.
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change in Z i is now derived by simultaneous solution of ( 1.3), (1.6), (1.9a), and ( 1. lo), with initial conditions T(0) = T,,,(O) = TR (1.11) Equation (1.11) implies that, at time zero, the system has been at rest at TR for some time, so that the thermocouple junction is at the same temperature as the tank. The solution to this system of equations is represented in Fig. 1.9 for a particular set of values of K, and K R. For this set of values, the effect of the thermocouple delay in transmission of the temperature to the controller is primarily to make the response somewhat more oscillatory than that shown in Fig. 1.7 for the same value of KR. However, if KR is increased somewhat over the value used in Fig. 1.9, the response is that shown in Fig. 1.10. The tank temperature oscillates with increasing amplitude and will continue to do so until the physical limitations of the heating system are reached. The control system has actually caused a deterioration in performance. Surely, the uncontrolled response for Kc = 0 in Fig. 1.4 is to be preferred over the unstable response of Fig. 1.10. This problem of stability of response will be one of our major concerns in this text for obvious reasons. At present, it is sufficient to note that extreme care must be exercised in specifying control systems. In the case considered, the proportional and integral control mechanism described by Eq. (1.9~) will perform satisfactorily if KR is kept lower than some particular value, as illustrated in Figs. 1.9 and 1.10. However, it is not difficult to construct examples of systems for which the addition of any amount of integral control will cause an unstable response. Since integral control usually has the desirable feature of eliminating steady-state error, as it did in Fig. 1.7, it is extremely important that we develop
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Zmk te.mpexatme versus time for increased KR.
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PROCESS SYSTEMS ANALYSIS AND CONTROL
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TR Comparator - Error N Controller ~tis~r% Heater
A l Heat input Tank -
Thermocouple 4 T FIGURE l-11
Block diagram for heated-tank system.
means for predicting the occurrence of unstable response in the design of any control system.
Block Diagram
A good overall picture of the relationships among variables in the heated-tank control system may be obtained by preparing a block diagram. This diagram, shown in Fig. 1.11, indicates the flow of information around the control system and the function of each part of the system. Much more will be said about blo ck diagrams in Chap. 9, but the reader can undoubtedly form a good intuitive notion about them by comparing Fig. 1.11 with the physical description of the process given in the previous paragraphs. Particularly significant is the fact that each component of the system is represented by a block, with little regard for the actual physical characteristics of the represented component (e.g., the tank or controller). The major interest is in (1) the relationship between the signals entering and leaving the block and (2) the manner in which information flows around the system. For example, TR and T,,, enter the comparator. Their difference, the error, leaves the comparator and enters the controller.
SUMMARY
We have had an overall look at a typical control problem and some of its ramifications. At present, the reader has been asked to accept the mathematical results on faith and to concentrate on obtaining a physical understanding of the transient behavior of the heated tank. We shall in the forthcoming chapters develop tools for determining the response of such systems. As this new material is presented, the reader may find it helpful to refer back to this chapter in order to place the material in proper perspective to the overall control problem.
PROBLEMS
1.1. Draw a block diagram for the control system generated when a human being sfeers an automobile.
THELAPLACE TRANSFORM
CHAPTER
THELAPLACE TRANSFORM
Even from our brief look at the control problem of Chap. 1, it is evident that solution of differential equations will be one of our major tasks. The Laplace transform method provides an efficient way to solve linear, ordinary, differential equations with constant coefficients. Because an important class of control problems reduces to the solution of such equations, the next three chapters are devoted to a study of Laplace transforms before resuming our investigation of control problems.
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