MECH466 - Lecture17-NyquistStabilityCriterion-2009W

5 05 40 2 10 1 10 0 10 1 10 100 2 10 2 10 0 0 0 10 1

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Unformatted text preview: 0 10 1 10 -100 -2 10 2 10 0 0 0 10 1 10 -50 2 10 0 0 -20 -1 10 -0.5 -40 -0.5 -100 -60 -150 -80 -2 10 2008/09 -1 10 0 10 1 10 2 10 -1 -1 -0.5 MECH466 : Automatic Control 0 0.5 1 7 -2 10 2008/09 -1 10 0 10 1 10 2 10 -1 -1 -0.5 MECH466 : Automatic Control 0 0.5 1 8 2 Example of Bode & Nyquist plots Nyquist stability criterion Third order system Third Bode plot Nyquist plot 1 0 Z: # of CL poles in open RHP Z: P: # of OL poles in open RHP (given) P: N: # of clockwise encirclement of -1 N: by Nyquist plot of OL transfer function L(s) L(s) (counted by using Nyquist plot of L(s)) L(s)) -50 0.5 -100 -150 -2 10 -1 10 0 10 1 2 10 10 0 0 -100 -0.5 -200 -2 -1 10 10 0 10 2008/09 1 2 10 10 -1 -1 Remark: N=-1: a counter-clockwise encirclement N=counter-0.5 0 0.5 1 MECH466 : Automatic Control 9 2008/09 MECH466 : Automatic Control Encirclements in Nyquist plot Clockwise Clockwise Remark Counter-clockwise CounterIm 10 If Nyquist plot passes the point -1, it means that If the closed-loop system has a pole on the closedimaginary axis (and thus, not stable). Im Im Re Re Re 2008/09 MECH466 : Automatic Control 11 2008/09 MECH466 : Automatic Control 12 3 Example for 2nd order L(s) Example for 3rd order L(s) Gain increase 1 1 Gain increase 15 1 10 0.5 0.5 0.5 5 0 0 0 -0.5 -0.5 -0.5 0 -5 -10 -1 -1.5 -1 -0.5 0 0.5 -1 1 -1.5 -1 -0.5 0 0.5 1 2nd For L(s) of order and constant numerator, L(s) gain increase never lead to unstable CL system! 2008/09 MECH466 : Automatic Control -1 -2 -1.5 -1 -0.5 0 0.5 1 -15 -5 0 5 10 15 For L(s) with relative degree 3, L(s) gain increase eventually lead to unstable CL system! 13 2008/09 MECH466 : Automatic Control 14 How to count # of encirclement Notes on Nyquist stability criterion A ray is drawn from -1 point in any convenient ray direction. Then, Nyquist stability criterion allows us to determine Nyquist the stability of CL system from a knowledge of the G(jω) of OL system. system. If an OL system is stable, it requires only If frequency response data of OL system (TF model L(s) is not necessary). L(s) It can deal with time delay, which Routh-Hurwitz It de...
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