Lecture 24- Review- Nyquist Plots, Nyquist Stability Criterion

Lecture 24- Review- Nyquist Plots, Nyquist Stability Criterion

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1 Lecture 24 ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 24: Stability Analysis for Feedback Control Systems Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 24 ECSE304 Signals and Systems II • The Nyquist Stability Criterion – The Encirclement property – Nyquist Contours – Nyquist Plots – Mapping of axis – Stability Criterion – Relates closed loop stability to open loop RHP zeros • Marginal Stability for Closed Loop Systems – Gain and Phase Margins Stability Analysis for Feedback Control Systems 12 3 456 7 s-plane () ZeroesEnc ENC PolesEnc NN N = j ω Enclosed pole Closed contour Nyquist Contour Nyquist Plot
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3 Lecture 24 ECSE304 Signals and Systems II Review: The Encirclement Property Encirclement Property : As a closed contour C in the s- plane is traversed once in the clockwise direction, the corresponding plot of G(s) for values of s along the contour encircles the origin in the clockwise direction a net number of times, , equal to the number of zeroes minus the number of poles contained within the contour ENC N 12 3 456 7 G(s) Plane s-plane ( ) ZeroesEnc ENC PolesEnc NN N = 4 Lecture 24 ECSE304 Signals and Systems II • Determine if has any RHP zeroes by defining the following closed contour: • The values of when plotted over this contour are known as the Nyquist plot. • Apply the encirclement property to determine if the closed loop transfer function is stable Review: Nyquist Plots As the contour encloses the entire RHP M →∞ 1( ) ( ) kK s H s + ) ( ) kK s H s + () () () ) ( ) kK s H s Ts kK s H s = +
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5 Lecture 24 ECSE304 Signals and Systems II • The values of along the semi-circle with radius reduce to a constant … assuming that there are at least as many poles as there are zeroes (n=m) Therefore, the Nyquist plot of can be obtained along the part of the contour that coincides with the imaginary axis Review: Nyquist Plots 1( ) ( ) kK s H s + 1 10 1 ) ( ) nn n s n bs b s b b kK s H s as a s a a →∞ ++ + += + " " M →∞ ) ( ) kK s H s + 6 Lecture 24 ECSE304 Signals and Systems II Review: Bode and Nyquist Plots • Second Order Example: [O and W, page 853] 20log () Hj ω H j ) 1 2 1 (1 ) 1 Hs ss = 40 dB per decade ( ) ( 180 deg) π ∞→− ) Magnitude and Phase Plots Nyquist Plot
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7 Lecture 24 ECSE304 Signals and Systems II • We have introduced the encirclement property to relate the number of poles and zeroes in a contour to the number of encirclements about the origin • We have introduced the Nyquist contour to relate the number of poles and zeroes in the RHP to the number of encirclements about the origin of the Nyquist plot • We will now derive the Nyquist criterion
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This note was uploaded on 02/08/2012 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Lecture 24- Review- Nyquist Plots, Nyquist Stability Criterion

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