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K, - 9 2Kc - 9 g+- s + -= o
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However, Aris and Amundson obtained the phase plane for (among other values) a value of K, slightly greater th& 9. This was accomplished by numerical solution of Eqs. (32.21). It was found that, in the vicinity of the steady-state point, the situation is as depicted in Fig. 32.10. There are two limit cycles surrounding the stable focus critical point. The inner limit cycle is unstable, and the outer limit cycle is stable, according to the definitions given earlier. It may be seen that any disturbance (or initial condition) which moves the system no further from the critical point than the unstable limit cycle can be controlled. That is, the control system will eventually bring the system back to steady state. However, once the system is forced outside this limit cycle, it will eventually spiral out to the stable limit cycle. Control cannot be restored, and the reactor temperature and concentration oscillate continuously. This example illustrates very well the limitations of linear control theory. All that the linear investigation could reveal is that, for K, > 9, the system will be stable in some vicinity of the control point. The phase-plane analysis shows that, for K, slightly greater than 9, this vicinity is inside the unstable limit cycle of Fig. 32.10. If K, is increased further, the two limit cycles disappear and good control can be achieved. This example points out the importance of unstable limit cycles. Although a physical system can never follow an unstable limit cycle, the limit cycle divides the phase plane into distinct dynamic regions for the physical system. Other values of K, were analyzed by Aris and Amundson. For low values of K,, there am two other critical points besides the control point. For example, for K, = 0.8, there are critical points at 8 = 1.77 y = 0.95 and 8 = 2.15 y = 0.15 Linear analysis shows that both these are stable, but for K, < 9, the control point (y = 0.5, 8 = 2) is not. Phase-plane analysis shows that, if the reactor is started at high temperatures, it will come to steady state at the high-temperature critical point and vice versa. Starting the reactor at the desired control point will be of no
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FIGURE 32-10 Stable and unstable limit cycles in exothetmic ical reactor.
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METHODS OF PHASE-PLANE ANALYSIS
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FIGURE 32-11 Phase-plane portrait of the control of a chemical reactor (limit cycle forms).
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avail, as it will leave and go to one of the other steady-state points, depending on the direction of the initial disturbance. For high values of K,, there is only one critical point, which is at the control point. Phase-plane analysis shows that Kc must exceed approximately 30 before rapid return to steady state at the desired control point, following all disturbances, is achieved. Some phase-plane portraits for this system that were obtained by means of a computer are shown in Figs. 32.11 to 32.13.
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FIGURE 32-12 Phase-plane portrait of the control of a chemical reactor (no limit cycle forms).
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!% i NONLINEAR CONTROL
FIGURE 32-13
Phase-plane of the control of a chemical reactor (no limit cycle forms).
This discussion is only a rather brief introduction to the extensive work by Aris and Amundson. The reader is strongly urged to consult the original paper for a more comprehensive treatment of the problem.
SUMMARY
We have seen that phase-plane analysis can be used for two typical nonlinear control problems. The results of this analysis give extensive information about the control system behavior. The responses to various disturbances can be visualized by sketching only a few trajectories. On the other hand, the method is effectively limited to second-order systems. Furthermore, analysis is considerably more laborious than the linear analysis, and a decision regarding the value of the additional information must be made.
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