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barcode reader using c#.net FlGURE 177 Block diagram for Example 17.1. in Software
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Drawing Bar Code In Visual C# Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Bar Code Generation In Java Using Barcode creation for Java Control to generate, create bar code image in Java applications. Since the closedloop system is secondorder, it can never be unstable. The shape of the response of the closedloop system to a unit step in R must resemble the curves of Fig. 8.2. The meaning of relative stability is illustrated by Fig. 8.2. The lower fi is made, the mom oscillatory and hence the less stable will be the response. Therefore, a relationship between phase margin and 62 will give the relation between phase margin and relative stability. To find this relation the openloop Bode diagram is prepared and is shown in Fig. 17.8. The simplest way to proceed from this diagram is as follows: consider a typical frequency w = 4. If the openloop gain were 1 at this frequency, then since the phase angle is 152 , the phase margin would be 28 . To make the openloop gain 1 ,at 0 = 4, it is required that Bar Code Decoder In Visual Studio .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications. Scanning Code 39 In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. &CL=
0.062 Then {*= 1 = 0.24 1 +Kc J
Hence, a point on the curve of 52 versus phase margin is l2 = 0.24 phase margin = 28
CONTROL SYSTEM
DESIGN BY
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Other points are calculated similarly at different frequencies, and the resulting curve is shown in Fig. 17.9. From this figure it is seen that 52 decreases with decreasing phase margin and that, if the phase margin is less than 30 , 52 is less than 0.26. From Fig. 8.2, it can be seen that the response of this system for 45 < 0.26 is highly oscillatory, hence relatively unstable, compared with a response for the system with phase margin 50 and 52 = 0.4. For the particular system of Example 17.1, it was shown that the response became more oscillatory as the phase margin was decreased. This result generalizes to mom complex systems. Thus, the phase margin is a useful design tool for application to systems of higher complexity, where the transient response cannot be easily determined and a plot such as Fig. 17.9 cannot be made. To repeat, the rule of thumb is that the phase margin must be greater than 30 . A similar statement can be made about the gain margin. As the gain margin is increased, the system response generally becomes less oscillatory, hence more stable. A control system designer will often try to make both the gain and phase margins equal to or greater than specified minimum values, typically 1.7 and 30 . Note that, for the caseof Example 17.1, the gain margin is always infinite because the phase lag never quite reaches 180 . However, the phase margin requirement of 30 necessitates that fi > 0.26, hence K, < 14, which means that an offset of 8 [see Eq. (17.2)] must be accepted. This illustrates the importance of considering both margins. The reader should refer to Fig. 17.6 to see that both margins exist simultaneously. Example 17.2. Specify the proportional gain Kc for the control system of Fig. 16.12. The Bode diagram for the particular case K, = 10 is presented in Fig. 16.13. The gain is to be specified for the two cases: 1. 70 = 0.5 min 2. 70 = 0 (no derivative action) 1 . Consider first the gain margin. The crossover frequency for the curve with derivative action is 8.0 rad/min. At this frequency, the openloop gain is 0.062 if the value of K, is unity. (Including the factor of l/l0 in the ordinate is actually FIGURE 179 Phase
margin
Damping versus phase margin for system of Fig. 17.7.
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equivalent to plotting the case K, = 1.) Therefore, according to the Bode criterion, the value of K, necessary to destabilize the loop is l/O.062 or 16. To achieve a gain margin of 1.7, K, must be taken as 1611.7 or 9.4. To achieve proper phase margin, note that the frequency for which the phase lag is 150 (phase margin is 30 ) is 5.3 rad/min. At this frequency, a value for K, of l/O.094 or 10.6 will cause the openloop gain to be unity. Since this is higher than 9.4, we use 9.4 as the design value of K,. The resulting phase margin is then 38O. 2. Proceeding exactly as in case 1 but using the curve in Fig. 16.13 for no derivative action, it is found that K, = 5.3 is needed for satisfactory gain margin and K, = 3.7 for satisfactory phase margin. Hence K, is taken as 3.7 and the resulting gain margin is 2.4. To see the advantage of adding derivative control in this case, note from Fig. 16.12 that the final value of C for a unitstep change in U is l/(l + Kc) for any value of 7~. The addition of the derivative action allows increase of the value of K, from 3.7 to 9.4 while maintaining approximately the same relative stability in terms of gain and phase margins. This reduces the offset from 21 percent of the change in U to 9.6 percent of the change in U. The reader is cautioned that the values of K, selected in this way should be regarded as initial approximations to the actual values,.which give optimal control of the system of Fig. 16.12. More will be said about this matter later in this chapter in conjunction with the twotank chemicalreactor control system of Chap. 11. Thus far, nothing has been said about upper limits on the gain and phase margins. Referring to Example 17.1 and Fig. 8.2, it is seen that, if (2 is too large, the response is sluggish. In fact, Fig. 8.2 suggests that for the system of Fig. 17.7 one should choose a value of 52 low enough to give a short rise time without causing excessive response time and overshoot. In other words, one wants the most rapid response that has sufficient relative stability. The results of Example 17.1 generalize to many systems of higher complexity, in terms of margin. Hence, the designer frequently chooses the controller so that either the gain or phase margin is equal to its lowest acceptable value and the other margin is (probably) above its lowest acceptable value. This was the procedure followed in Example 17.2. In almost every situation, the designer faces this conflict between speed of response and degree of oscillation. In addition, if integral action is not used, the amount of the offset must be considered. The concepts of gain and phase margin are useful in selecting K, for proportional action. However, for additional modes of control such as PD, these concepts are difficult to apply in practice. Consider the selection of K, and 70 in Example 17.2. For a different value of 70 the derivative contribution is shifted to the right or left on the Bode diagram of Fig. 16.13. This means that a different value of K, will provide the proper margins. A typical design procedure is to select the value of 70 for which the value of K, resulting in a 30 phase margin is maximized. The motivation for this choice is that the offset will be minimized. However, the procedure is clearly trial and error. In the case of threemode control, there are two parameters, ~1 and 70, which must be varied by trial to meet various design

