Closed-loop stability sensitivity

We call this high sensitivity of the recycle flowrates to small disturbances the snowball effect. We illustrate its occurrence in the simple example below, It is important to note that this is not a dynamic effect it is a steady-state phenomenon. But it does have dynamic implications for disturbance propagation and for inventory control. It has nothing to do with closed-loop stability. However, this does not imply that it is independent of the plant s control structure. On the contrary, the extent of the snowball effect is very strongly dependent upon the control structure used. 

Example 11—Sensitivity of closed-loop stability to small variations in controller parameters. For the stable transfer function... 

We have chosen the steady state with Yfa = 0.872 and FCD = 1.0 giving a dense phase reactor temperature of Yrd = 1.5627 (Figure 7.14(b) and (c)) and a dense-phase gasoline yield of x-id = 0.387 (Figure 7.14(a)). This is the steady state around which we will concentrate most of our dynamic analysis for both the open-loop and closed-loop control system. We first discuss the effect of numerical sensitivity on the results. Then we address the problem of stabilizing the middle (desirable, but unstable) steady state using a switching policy, as well as a simple proportional feedback control. 

Frequency response techniques are powerful tools for the design and analysis of feedback control systems. The Bode and Nyquist stability criteria provide exact stability results for a wide variety of control problems, including processes with time delays. They also provide convenient measures of relative stability, such as gain and phase margins. Closed-loop frequency response concepts such as sensitivity functions and bandwidth can be used to characterize closed-loop performance. 

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