Control Systems. Frequency Method Nyquist Analysis.

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1 Frequency Method Nyquist Analysis

2 Outline Polar plots Nyquist plots Factors of polar plots

3 PolarNyquist Plots Polar plot: he locus of the magnitude of ω vs. the phase of ω on polar plane as ω goes from 0 to ood: it depicts the frequency response characteristics of a system over entire frequency range in a single plot Bad: the plot does not clearly indicate the contributions of each individual factors of the open-loop transfer function

4 Integrator Integrator Polar plot negative imaginary axis 270 s s 0 / tan

5 Derivative Derivative s s 90 tan 0 Polar plot positive imaginary axis

6 st Order Systems st order system Recall Bode plot s s 2 2 tan 2 2 tan 90 0, For 45, 2 For 0, For

7 st Order Zero st order zero 2 2 tan 2 2 tan 90, For 45 2, For 0, For

8 Basic factors: 2 nd order system 2 nd order system Frequency point whose distance from the origin is a maximum the resonant frequency n n 80 0, For 90, 2 For 0, For n n n

9 Basic factors: 2 nd order zeros 2 nd order zeros 2 n n 2 0

10 eneral Polar Plots l = 0 ype 0 Plot starts on the positive real axis with a tangent perpendicular to the real axis. erminal point ω= is at the origin Curve is tangent to one of the axis which one depends on relative degree, odd gives ω-axis 2 n n n n m m m m b a a a b b K l

11 eneral Shape of Polar Plots l = ype ω = 0 Plot starts at infinity with angle -90, parallel to Im ω = Plot ends at the origin tangent to one of the axis l = 2 ype 2 ω = 0 Plot starts at infinity with angle -80, parallel to -Re ω = Plot ends at the origin tangent to one of the axis Every free integrator adds 90 o of phase and rotates low frequency portion of Nyquist plot

12 eneral Polar Plots Arrival angle to origin: determined by n-m 2 n n n n m m m m b a a a b b K l

13 Polar Plots of Standard ransfer Functions

14 Polar Plots of Standard ransfer Functions

15 Outline Mapping of complex function Cauchy theorem Nyquist stability criterion

16 Motivation Nyquist stability criterion: a graphical technique for determining the stability of a system. based on Cauchy theorem on functions of a complex variable only need the polar plot of the open loop systems # of each type of right-half-plane singularities must be known. can be applied to systems defined by non-rational functions, such as systems with delays. restricted to linear, time-invariant systems.

17 Complex Mapping in the s-plane For a complex function Fs, any point in the s-plane can be represented in the Fs plane as long as s isn t a pole of Fs Ex: Im F s 2s 2 Im Re Re s - plane Fs - plane

18 Complex Mapping in the s-plane A contour drawn in the complex s plane, encompassing but not passing through any number of zeros and poles of a function, can be mapped to another plane Fs pla ne by the function Fs. F s 2s Conformal mapping angle preserved

19 Complex Mapping in the s-plane Complex rational function F s s s 2 Conformal mapping angle preserved he area enclosed by a contour is the area to the right as the contour is traversed in the clockwise direction

20 Cauchy s heorem If a contour in the s-plane encircles Z zeros and P poles of Fs and does not pass through any poles or zeros of Fs and the traversal is in the clockwise direction along the contour, the corresponding contour in the Fs-plane encircles the origin of t he Fs-plane N=Z-P times in the CW F s s s 0.5 N Z P 0

21 Cauchy s heorem Ex. Fs N Z P 3 2

22 Cauchy s heorem Fs N Z P 0

23 Concept of Nyquist Stability Criterion Use Cauchy heorem to determine stability Draw Nyquist contour that encircles the entire RHP Contour goes along axis, then circles back with infinite-radius halfcircle If any poles or zeros are in the RHP, they show up as encirclements of Fs at origin assume no axis poles for now

24 Concept of Nyquist Stability Criterion Let Fs=+sHs For the system to be stable, all zeros or roots of Fs must lie in LHP. he number of unstable zeros of Fs is thus Z N P When the system is open-loop stablep=0, then Z=N

25 Nyquist Stability Criterion Rs + - Es s Ys Hs Examining stability of Fs=+sHs is the same as examining the CW encirclements of -+0 by shs contour

26 Nyquist Stability Criterion Rs + - Es s Ys Hs Assume n-m 0. If shs has k poles in the RHP and then, as goes from - to +, H must encircle -+0 k times in the CW direction for stability

27 Use of Nyquist Stability Criterion In summary Z N P Z : # of zeros of +shs in RHP i.e Closed-loop poles P : # of poles of shs in RHP Open-loop poles N : # of CW encirclements of the -+0 by shs For stable system Z=0, Open-loop stable plant: P=0 N=0 A feedback system is stable if and only if the contour in shs plane does not encircle the -,0 point. Open-loop unstable plant: P 0 N=-P A feedback system is stable if and only if, for the contour in shs plane, the number of counterclockwise of the -,0 point is equal to the number P of poles of with positive real parts.

28 Use of Nyquist Stability Criterion Z N Z : # of zeros of +shs in RHP i.e Closed-loop poles P : # of poles of shs in RHP Open-loop poles N : # of CW encirclements of the -+0 by shs P Following scenarios possible No encirclement of - System is stable if there are no poles of shs in RHP Otherwise unstable CCW encirclement of - System is stable if # of CCW encirclements = # poles of shs in RHP Otherwise unstable CW encirclement of - Unstable system

29 st order system Ex: s H s s Nyquist plot Rs + - Es s Hs Ys 0.5 Nyquist Diagram Imaginary Axis 0 # encirclements : N = 0 # of Poles in RHP: P = 0 Z = N + P = 0 stable Real Axis

30 Unstable system Ex: s 2 s Nyquist Diagram Imaginary Axis # CW encirclements: N = - # of Poles in RHP: P = Z = N + P = 0 stable Real Axis If gain 2 is reduced, N becomes 0 and system becomes unstable

31 2 nd order system Ex: s H s s 00 s 0 Rs + - Es s Hs Ys Nyquist plot # encirclements: N = 0 # of Poles in RHP: P = 0 Z = N + P = 0 stable

32 Poles/Zeros on the -Axis For case with -axis poles? Nyquist path must not pass through poles or zeros of shs use a semicircle with the infinitesimal radius K e... K K s e e H e e e e... e

33 Pole on the -Axis Ex: system w/ a pole at origin s H s K s s +90 degrees at A =0-0 degrees at B -90 degrees at C =0 + # encirclements : N = 0 # of Poles in RHP: P = 0 Z = N + P = 0 stable

34 Example Ex: s s s Nyquist plot 5 Nyquist Diagram 0 Imaginary Axis Real Axis # encirclements: N = 0 # of Poles in RHP: P = 0 Z = N + P = 0 stable

35 Example Ex: s K s s 2s Nyquist plot

36 Example Ex: s K s s 2 = 2 = for previous example Nyquist plot K= K=2 K=3

37 Example Ex: system w/ 2 poles at origin s 2 s K s Nyquist plot # encirclements : N = 2 # of Poles in RHP: P = 0 Z = N + P = 2 unstable

38 Example Ex: system w/ a pole at RHP s K s s Nyquist plot # encirclements : N = # of Poles in RHP: P = Z = N + P = 2 unstable

39 Plots for ypical ransfer Functions

40 Plots for ypical ransfer Functions

41 Plots for ypical ransfer Functions

42 Plots for ypical ransfer Functions

43 Plots for ypical ransfer Functions

44 Plots for ypical ransfer Functions

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