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9 - nyquist9 filled - ME 575 Handouts Nyquist Stability...

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Unformatted text preview: ME 575 Handouts Nyquist Stability Controller R(s) + − Plant C G let L(s) = CG = Y(s) NC (s)NG (s) NL (s) = ; DC (s)DG (s) DL (s) characteristic eq: 1 + L(s) = 0 let F(s) = 1 + ME575 Session 9 – Nyquist Stability NL (s) DL (s) + NL (s) = =0 DL (s) DL (s) School of Mechanical Engineering Purdue University Slide 1 Stability in Freq. Domain • We want to know where the zeros of F(s) are. • Use a MAPPING between the s-plane (where the roots are) and the F(s)-plane. • Since we want stability, isolate the RHP with a geometrically simple directed contour Γs. ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 2 1 ME 575 Handouts Effect of Zeros of F(s) 1) Zero outside contour: F(s) = s + a, a > 0 Im s Im F Re s ME575 Session 9 – Nyquist Stability ⇒ Re F School of Mechanical Engineering Purdue University Slide 3 Effect of Zeros of F(s) 2) Zero inside contour: F(s) = s − a, a > 0 Im s Im F Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering Purdue University Re F Slide 4 2 ME 575 Handouts Principle of the Argument (Cauchy) Let F(s) be a single-valued function that has a finite number of poles in the s-plane. Choose a closed path Γs in the s-plane such that it avoids any poles or zeros of F(s). Then the corresponding contour ΓF mapped in the F(s)plane will encircle the origin NCW times in a clockwise direction. ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 5 Application of P of A to Stability Recall: R(s) + L(s) = CG N (s) =L DL (s) Controller − let F(s) = 1 + L(s) = 1 + C Plant G Y(s) NL (s) DL (s) + NL (s) = =0 DL (s) DL (s) zeros of F(s) poles of F(s) ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 6 3 ME 575 Handouts Relate to Loop Transfer Function F(s) = 1 + L(s) ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 7 Nyquist Stability Criterion NCW = NZ − NP NCW = net # of CW encirclements of −1 by ΓL NZ = # of closed-loop poles encircled by ΓS NP = # of open-loop poles encircled by ΓS A feedback system having NP open-loop poles in the RHP is stable if and only if the Nyquist plot of L(s) encircles −1 NP times in a counterclockwise direction. ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 8 4 ME 575 Handouts Steps in Sketching a Nyquist Plot • Plot poles of L(s) in the s-plane. • Draw the Nyquist contour Γs, indenting to the right of any poles of L(s) on the imaginary axis. • Map contour Γs to L(s)-plane. • Apply encirclement condition. School of Mechanical Engineering Purdue University ME575 Session 9 – Nyquist Stability Slide 9 Nyquist Example L(s) = 2K (2s + 1)(s + 1)(s / 2 + 1) Im s Im L Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering Purdue University Re L Slide 10 5 ME 575 Handouts Nyquist Plot ① s = jω: L( jω) = L( jω) = 2 (1 + j2ω)(1 + jω)(1 + jω / 2) 2 1 + (2ω)2 1 + ω2 1 + (ω / 2)2 ∠L( jω) = − tan−1 2ω − tan−1 ω − tan−1 ω / 2 ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 11 School of Mechanical Engineering Purdue University Slide 12 Nyquist Plot ② s = Rejφ: ③ s = −jω: ME575 Session 9 – Nyquist Stability 6 ME 575 Handouts Nyquist Stability Criterion School of Mechanical Engineering Purdue University ME575 Session 9 – Nyquist Stability Slide 13 Nyquist Plot for Arbitrary K Im s Im L Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering Purdue University Re L Slide 14 7 ME 575 Handouts Relative Stability Proximity to encirclement of −1 is a measure of closeness to instability for the nominal system, i.e., relative stability. Relative stability is quantified via: Gain Margin – the factor by which the open-loop gain opencan be increased at a phase of −180° before the 180° system goes unstable. Phase Margin – the amount by which open-loop phase opencan be decreased at unity magnitude before system goes unstable. Sensitivity Peak ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 15 Margins and the Nyquist Plot Lo ( s ) = ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University 3 (s + 1)3 Slide 16 8 ME 575 Handouts Margins and Bode Plots Lo ( s ) = 3 (s + 1)3 |Lo(jωgc)| = 1 ∠Lo(jωpc) = −180° ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 17 Summary of Margins • For stability, we want no encirclement of −1 (for minimum-phase systems): GM > 1 or GMdB > 0 PM > 0° • As measures of relative stability, more positive GM & PM imply farther away from instability: GM indicates allowable extra gain PM indicates allowable extra phase lag (time delay) ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University Slide 18 9 ME 575 Handouts Sensitivity Peak Lo ( s ) = 3 (s + 1)3 1 + Lo ( jω) > η So ( jω) = = ME575 Session 9 – Nyquist Stability School of Mechanical Engineering Purdue University 1 1 < 1 + Lo ( jω) η Slide 19 10 ...
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This note was uploaded on 12/09/2011 for the course ME 575 taught by Professor Meckl during the Fall '10 term at Purdue.

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