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**Example text**

G in the reference-output equation is treated as a variable to determine this sensitivity. dyG ð3:1Þ S¼ r y dG r The first term in this product is the derivative of the reference-output equation with respect to variable G. d yr d CG ¼ ð3:2Þ dG dG 1 þ CG C ð3:3Þ ¼ ð1 þ CGÞ2 The sensitivity is C Gð1 þ CGÞ S¼ 2 CG ð1 þ CGÞ ð3:4Þ ¼ 1 1 þ CG ð3:5Þ ¼ 1 1þT ð3:6Þ It is evident that the closed-loop system is insensitive to plant parameter variations at frequencies where the feedback is large. g. the inclusion or extraction of poles and zeros), rather those that effect small changes in the plant gain.

On the other hand, a few decibels of performance improvement may be the difference between the system performing to required specifications or failing to (military) or dominating the competition or languishing behind (commercial). Having established the positive aspects of feedback, practical limitations to the available quantity are now presented. 3 Bode sensitivity integral For a system with greater than first-order roll-off and Np open-loop right half plane (RHP) poles pi, i ¼ 1, 2, . . Np, the sensitivity of the closed-loop system satisfies the following (see Appendix A): p Np X Re½pi ¼ ð1 logjSðjwÞjdw ð3:7Þ 0 i¼1 This relationship gives great insight into the limitations of control performance for systems with greater than first-order roll-off.

Definition: Functional bandwidth For a low pass loop transmission and |T(jw)| is approximately constant for w < wf, wf is the functional bandwidth. Definition: Negative feedback Feedback is negative at frequencies where |F(s)| > 1. It is noted that many textbooks refer to negative feedback as a result of the negative sign on the reference/ feedback summing junction. While loop phase is critical, there is nothing special about the sign at this summing junction. For instance, if a À1 block is placed in the feedback path, then the summing junction could have plus signs at both inputs with no change to the closed-loop system.