MIT6_241JS11_lec16

MIT6_241JS11_lec16 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 16: Bode’s Sensitivity Integral Readings: DDV, Chapter 18 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 4, 2011 E. Frazzoli (MIT) Lecture 16: Bode Integral April 4, 2011 1 / 7
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Cauchy’s integral theorem Let Ω C be an open , simply connected set. Let f : Ω C be a holomorphic function. In other words, the limit f ( s 0 ) = lim f ( s ) f ( s 0 ) s s 0 s s 0 exists (and is continuous) for all s 0 Ω. Note that a complex function is holomorphic if and only if it is analytic. Let γ : [0 , 1] Ω be a differentiable function, such that γ (0) = γ (1). Then, 1 f ( γ ( t )) γ ( t ) dt = f ( z ) dz = 0 , 0 Γ where Γ is the closed contour traced by γ ( t ) as t ranges from 0 to 1. E. Frazzoli (MIT) Lecture 16: Bode Integral April 4, 2011 2 / 7
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± ± ± ± ± ± ± ± Cauchy’s integral formula Let Ω, f , γ , and Γ be as in the previous slide. Furthermore,
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This note was uploaded on 01/12/2012 for the course ECE 6.241j taught by Professor Prof.emiliofrazzoli during the Spring '11 term at MIT.

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MIT6_241JS11_lec16 - 6.241 Dynamic Systems and Control...

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