barcode reader integration with asp.net Typical control loop containing nonlinear element. in Software

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Phase FIGURE 33.5 Gain-phase plot for system of Fig. 33.1.
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The advantages of the gain-phase plot are (1) elimination of trial-and-error solution of equations such as Eqs. (33.10) and (2) ease of treatment of complex linear systems G,H. In addition, the gain-phase plot can be used to estimate the occurrence or nonoccurrence of a limit cycle, according to whether or not an intersection occurs.
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The describing function can be used to good advantage for estimation of amplitude and frequency of limit cycles in systems similar to the one studied here. The success of the method depends on the presence of a sufficient number of linear elements in the loop to filter out the harmonics generated by the nonlinear element. No information about the transient response is obtained. However, the method requires considerably less labor than does the phase-plane method, and the limit cycle amplitude and frequency are often the quantities of primary interest. It should also be noted that the describing function method is not limited to second-order systems, as is the phase-plane method. In fact, the higher the order of G,,H in Fig. 33.4, the more accurate will be the describing function results.
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33.1. For the control system shown in Fig. P33.1 determine the frequency and amplitude of the limit cycle if one exists. Use the describing function method.
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FIGURE P33-1
33.2. (a) For the system shown in Fig. P33.2, r = I,5 = 1/2, K = 1, M = 1, b = 0.25 and R = 0. (b) Show that the describing function is
where A is the amplitude of the sine wave entering the nonlinearity. (c) For K = 2, does a limit cycle exist If so, describe it. (d) If a transport lag ems is introduced in the feedback loop, determine if a limit cycle exists for K = 2.
K 2w + 2( S + 1 ,
FIGURE P33-2
33.3. The stirred-tank system shown in Fig. P33.3 produces an aqueous solution of salt by use of a solenoid valve that switches from one reagent tank to the other as described below. The reagent tanks contain concentrated solutions of salt. When the measured concentration is above the set point, the control reagent of lower concentration enters the mixing tank at a constant flow rate of 0.01 liter/mitt. When the measured concentration is less than the set point, the control reagent of higher concentration enters the mixing tank at a constant rate of 0.01 liter/min. The hold-up volume of the tank is 2 liters, the transport lag between the tank and measuring element is 1.2 min, and the set point is 2 g salt/liter. (a) Obtain a block diagram, in terms of deviation variables, for this control system. (b) By means of the describing function method, determine the characteristics of the limit cycle (frequency and amplitude), if one exists.
THE DESCRIFXNG
FUNCTION TECHNIQUE
-- et - - - -- 1 0 0g s a l t / liter
Control
reagent \ _- wz - - -
300 g salt/
liter
FIGURE P33-3
33.4. For the control system shown in Fig. P33.4, determine if a limit cycle exists for K = 1, 2, and 3. If a limit cycle exists, describe it in terms of amplitude and frequency. For the nonlinearity shown, N = I[ sin- (i)+ i,/G] N = 1 forA< 1 ,$OforAZ 1
A is the amplitude of the sine wave entering the nonlinearity
FIGURE P33-4
PART
COMPUTERS INPROCESS CONTROL
CHAPTER
DIGITAL COMPUTER SIMULATION OFCONTROL SYSTEMS
The purpose of this chapter is to describe some of the methods for obtaining the transient response of a control system from a set of differential equations or transfer functions. Inversion of a high-order transfer function can be a time-consuming task. If a control system includes a nonlinearity or a transport lag, obtaining the response as an analytical expression is often impossible. The appearance of analog computers and digital computers after World War II made the task of solving the dynamic response of control systems much easier. During the period from the mid-fifties to the mid-seventies, the analog computer was widely used to obtain the response of control systems. During that time, digital computing was very costly and slow compared to the situation today. There was little software available; at the beginning, one had to program the solution to a problem using machine code. The basic elements of the analog computer consisted of integrators, summers, gain potentiometers, and some nonlinear devices, such as multipliers. By connecting these computing elements together with wires, one could obtain the transient response to a rather large-scale control problem in the form of a voltage that varied with time. As the cost of digital computing decreased and its speed of operation increased, the analog computer was gradually replaced with the digital computer. This change was especially noticed with the availability of the personal computer. 517
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