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Extra info for Selective Guide to Literature on Integrated Circuits (Engineering Literature Guides, No. 18)

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In a purely mathematical sense most of the formulae are related to one another by such obvious transformations and changes of variable that there is no good reason for picking out any particular set as independent. Basically they are merely reflections of Cauchy’s theorem. 2 Bode Integrals for S and T This subsection starts this book’s main journey through results concerning complementary mappings, by which we mean mappings relating signals of the loop and such that their sum is a constant mapping.

3 that S and T are indicators of feedback system performance. We will next consider how these functions are affected by plant variations. We focus on the scalar case but similar conclusions hold, mutatis mutandis, for multivariable systems. Assume that the loop gain changes from its nominal value L to its actual ˜ It is not difficult to show that the relative changes in the sensitivity value L. and complementary sensitivity functions are given by L˜ − L S˜ − S = −T , L S˜ T˜ − T L˜ − L =S . T˜ L˜ These relations show that the sensitivity function will be robust with respect to changes in the loop gain in those frequency ranges where the nominal complementary sensitivity is small (typically at high frequencies); conversely the complementary sensitivity will be robust with respect to changes in the loop gain in those frequency ranges where the nominal sensitivity is small (typically at low frequencies).

For one flight condition, the unstable pole is at 6 and the nonminimum phase zero at 26. It is desired to use a stable, minimum phase controller in series with G, such that the closed loop is stable and has a phase margin of π/4. Consider the open-loop system formed by the cascade of plant and controller and modeled by a transfer function, L say. This system is neither stable nor minimum phase in open loop. 37) ˜ where L˜ is stable, minimum phase, and such that L(0) > 0, and where p and q correspond to the (real) ORHP pole and zero of G respectively.

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