Bode stability criterion transfer functions

All systems in Example 18.1 have an important common feature The AR and of the corresponding open-loop transfer functions decrease continuously as co increases. This is also true for the large majority of chemical processing systems. For such systems the Bode stability criterion leads to rigorous conclusions. Thus it constitutes a very useful tool for the stability analysis of most control systems of interest to a chemical engineer. 

IV.58 Using the Bode stability criterion, find which of the control systems with the following open-loop transfer functions are stable and which are unstable ... 

IV.59 Consider the processes with the transfer functions given in Problem IV.23. Each of these processes is feedback controlled with a proportional controller. Assume that Gm = Gf = 1. Using the Bode stability criterion, find the range of values of the proportional gain Kc which produce stable (if it is possible) closed-loop responses. 

Next we state one of the most important results of frequency response analysis, the Bode stability criterion. It allows the stability of a closed-loop system to be determined from the open-loop transfer function. 

The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for GqiXs), a polar plot of its frequency response characteristics (see Chapter 14). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of co or cOg. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. 

At this point, it is appropriate to summarize the relative advantages and disadvantages of the Bode and Nyquist plots. The Bode plot provides more information than the Nyquist plot, because the frequency is shown explicitly. In addition, it facihtates analysis over a wide range of frequencies due to its logarithmic frequency scale. Another advantage of the Bode plot is that it allows the open-loop frequency response characteristic to be graphically constructed from the characteristics for the individual transfer functions, G, Gy, Gp, and G, as shown in Chapter 14. The chief advantage of the Nyquist plot is that the Nyquist stability criterion is more widely applicable than the Bode stability criterion. 

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