barcode reader using c#.net FIGURE 17-4 Phase behavior of complex system for which Bode criterion is not applicable. in Software

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FIGURE 17-4 Phase behavior of complex system for which Bode criterion is not applicable.
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FIGURE 17-5 Block diagram for general control system.
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Application of the criterion requires nothing more than plotting the openloop frequency response. This may be based on the theoretical transfer function, if it is available, as we have done for the tank-temperature system. If the theoretical system dynamics are not known, the frequency response may be obtained experimentally. To do this, the open-loop system is disturbed with a sine-wave input at several frequencies. At each frequency, records of the input and output waves are compared to establish AR and phase lag. The results ate plotted as a Bode diagram. This experimental technique will be illustrated in more detail in Chap. 19. For the remainder of this chapter, we accept the Bode stability criterion as valid and use it to establish control system design procedure.
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Gain and Phase Margins
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Let us consider the general problem of selecting G Js) for the system of Fig. 17.5. Suppose the open-loop frequency response, when a particular controller G,(s) is tried, is as shown in the Bode diagram of Fig. 17.6. The crossover frequency, at which the phase lag is 1 80 , is noted as o,, on the Bode diagram. At this frequency, the AR is A. If A exceeds unity, we know from the Bode criterion that the system is unstable and that we have made a poor selection of G,(S). In Fig. 17.6 it is assumed that A is less than unity and therefore the system is stable. It is necessary to ascertain to what degree the system is stable. Intuitively, if A is only slightly less than unity the system is almost unstable and may be expected to behave in a highly oscillatory manner even though it is theoretically stable Furthermore, the constant A is determined by physical parameters of the system, such as time constants. These can be only estimated and may actually change slowly with time because of wear or corrosion. Hence, a design for which A is close to unity does not have an adequate safety factor. To assign some quantitative measure to these considerations, the concept of gain margin is introduced. Using the nomenclature of Fig. 17.6, Gain margin = i
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* Again, heuristic arguments are used. This statement is self-evident to the reader who has studied Chap. 15, where it is shown that the roots of the characteristic equation vary continuously WI&~ system parameters. Proof of the statement requires the Nyquist stability criterion.
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Open-loop Bode diagram for typical control system.
ljrpical specifications for design are that the gain margin should be greater than 1.7. This means that the AR at crossover could increase by a factor of 1.7 over the design value befom the system became unstable. The design value of the gain margin is really a safety factor. As such, its value varies considerably with the application and designer. A gain margin of unity or less indicates an unstable system. Another margin frequently used for design is the phase margin. As indicated in Fig. 17.6, it is the difference between 180 and the phase lag at the frequency for which the gain is unity. The phase margin therefore represents the additional amount of phase lag required to destabilize the system, just as the gain margin represents the additional gain for destabilization. Typical design specijications are that the phase margin must be greater than 30 . A negative phase margin indicates an unstable system.
Example 17.1. Find a relation between relative stability and the phase margin for the control system of Fig. 17.7. A proportional controller is to be used. This block diagram corresponds to the stirred-tank heater system, for which the block diagmm has been given in Fig. 13.6. The particular set of constants is T=7m=l 1 -cl
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