9 - nyquist9 filled

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

• Fall '10
• Meckl
• Trigraph, Purdue University, Nyquist plot, Nyquist stability criterion, Electrical parameters, School of Mechanical Engineering

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern