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34.1. The following differential equation is to be solved by digital computation: - = 2t - 1.5y dt shown Y(O) = 0
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A portion of a computer program, which uses the Runge-Kutta method, is below: 25 DY (T,Y) = 2*T - lS*Y 26 DT = 0.1 27 Y = 0.0 28 T = 0.0 29 Kl = DY (T,Y)*DT 30 K2 = DY (T + DT*.5, Y + Kl*S)*DT 31 K3 = DY (T + DT*.5, Y + K2*.5)*DT 32 K4 = DY (T + DT*.5, Y + K3*.5)*DT 33 Y = Y + (Kl + 2*K2 + 2*K3 + K4)*DT/6 34T=T+AT 35 PRINT T,Y 36 IF (T.LT.2.) GO TO 28 37 STOP 38 END
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(a) Indicate any errors you find in this program by noting the statement number of the line where it appears; also describe the error and correct it if you can. (b) Do one cycle of calculation by hand using the Runge-Kutta method and obtain the value of K1, K2, K3, and K4 for use in getting Y at t = 0.1. (c) Also obtain Y at I = 0.1 by using the Runge-Kutta method. 34.2. The control system shown in Fig. P34.2 is to be simulated by digital computation. A portion of a computer program, which uses the Runge-Kutta method, is shown below. 24 DY (T,Y) = (1 + KC)*DT - KC*Y 25 KC = 2.0 26 DT = 0.1 27 Y = 1.0 28 T = 0.0 29 Kl = DY(T,Y)*DT 30 K2 = DY(T + DT, Y + Kl)*DT 31 K3 = DY(T + DT*.5, Y + K2*.5)*DT 32 K4 = DY(T + DT*.S, Y + K3*.5)*DT 33 Y = Y + (Kl + 2*K2 + 2*K3+K4)*DT
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FIGURE P34-2
34 T = T +AT 3 5 PRINT T,Y 36 IF (T.LT.2.) GO T O 27 3 7 STOP 3 8 END (a) Indicate any errors you find in this program by noting the statement number of the line where it appears; also describe the error and correct it it you can. (b) After correcting the program, do one cycle of calculation by hand using the Runge-Kutta method and obtain the value of Kl, K2, K3, and K4 for use in getting Y at t = 0.1. 34.3. The step response of the following differential equation is to be-obtained numerically with the aid of a digital computer.
$$+0.8$+y
dyldt = 0 and y = 0 at t = 0 Integration step sizes (At) of 0.1, 0.5, and 1.0 are to be used. (a) Which of the step sizes will give a numerical solution closest to the analytical solution (b) Which step size will require the least computation time (c) If it is possible to get an impulse response for the above differential equation, show how you would provide for it in solving the differential equation by the computer. 34.4. Write a computer program in BASIC to simulate the response of the PID control system of Example 34.4 for a unit-step change in load (U = l/s) for the case of K, = 2.0, q = 1, Td = 1, and T = 2.
CHAPTER
MICROPROCESSOR-BASED CONTROLLERS AND DISTRIBUTED CONTROL
In this chapter, some of the highlights of modern industrial microprocessor-based controllers and distributed control systems will be presented. A microprocessorbased controller is essentially a digital computer programmed to perform the function of a process controller. For our purpose, the term microprocessor is synonymous with computer and we could refer to a microprocessor-based controller as a computer-based controller. The number of features of these modern controllers is far too great to cover in one chapter. The best way for the reader to acquire some experience with modern controllers is through laboratory and plant use and by attending some of the short courses offered by the major suppliers of the equipment. HISTORICAL RACKGROUND During the past fifty years tremendous development has occurred in process control hardware. The three phases of development am pneumatic control, electronic control, and microprocessor-based control. During the 1940s the predominant controller was pneumatic, meaning that signals to and from the controller and within the controller mechanism were air-pressure signals that usually varied from 3 to 15 psig. The development of the high-gain operational electronic amplifier during the second world war led to the development of the electronic controller and also the analog computer. The electronic controller mimicked the control functions of the pneumatic controller. It also provided some improvements, such as accurate and reproducible control parameter settings and reduction in size of the instruments. In contrast, the pneumatic controller required frequent calibration of the knobs used to set the various controller parameters (K,, ~1, rn). The pneu543
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