MECH466 - Lecture17-NyquistStabilityCriterion-2009W

System if an ol system is stable it requires only if

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Unformatted text preview: lay, Routhcriterion cannot. We often draw only half of Nyquist plot. (The We other half is mirror image w.r.t. real axis.) w.r.t. Im B Count N for cases: A = -1 B = -1 A Re Re 2008/09 MECH466 : Automatic Control 15 2008/09 MECH466 : Automatic Control 16 4 Some examples Example for unstable L(s) Unstable L(s) Unstable L(s) Stable L(s) Stable L(s) L(s) with an integrator L(s) L(s) with a double integrator L(s) L(s) with a time-delay L(s) time- 0.1 0.05 0 -0.05 -0.1 2008/09 MECH466 : Automatic Control 17 2008/09 Example of unstable L(s) (cont’d) -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 MECH466 : Automatic Control 18 Interpretation by root locus Im 0.15 Gain increase 0.06 0.1 Gain decrease Destabilizing! 0.04 0.05 0.02 0 -0.05 Re 0 -0.02 -3 -2 -0.1 -0.06 -2 2008/09 1 -0.04 -1.5 -1 -0.5 0 -1 MECH466 : Automatic Control -0.8 -0.6 -0.4 -0.2 0 19 Open loop gain increase will first stabilize, and then, destabilize the closed-loop system. closed2008/09 MECH466 : Automatic Control Stabilizing! 20 5 Nyquist criterion: A special case Example for stable L(s) 0.5 0 0 IF P=0 (i.e., if L(s) has no pole in open RHP) IF L(s) -0.5 -5 -1 -1.5 -10 -2 -2.5 This fact is very important since open-loop systems openin many practical problems have no pole in open RHP! 2008/09 MECH466 : Automatic Control -3 21 Interpretation by root locus CL stable -1 2008/09 0 1 2 CL unstable -15 -5 3 0 5 10 15 MECH466 : Automatic Control 20 22 Example of L(s) with an integrator We modify Nyquist path: We Im Im Re (very small) s Re -3 -2 -1 (very large) Open loop gain increase will destabilize the closed-loop system. closed2008/09 MECH466 : Automatic Control 23 2008/09 MECH466 : Automatic Control 24 6 Example of L(s) with an integrator L(s) for modified Nyquist path For small |s|, For Im 15 s Re (very small) 10 When s moves as When L(s) moves as L(s) 5 0 L(s) L(s) -5 -10 Note: If L(s) has no open RHP pole, L(s) we are interested in (and draw) Nyquist plot ONLY around the critical point -1. -15 -1 2008/09 -0.8 -0.6 -0.4 -0.2 0 MECH466 : Automatic Control 25 2008/09 MECH466 : Automatic Control 26 L(s) with a time-delay Example of L(s) with double integrator 0.5 5 0 -0.5 -1 -1.5 0 -2 -2.5 -3 -5 -5 2008/09 -4 -3 MECH466 : Automatic Control -2 -1 0 -1 0 1 2 3 Routh-Hurwitz is NOT applicable! Routh27 2008/09 MECH466 : Automatic Control 28 7 Bode plot of a time delay (review) 1 0.5 0 -0 . 5 -1 -2 10 10 -1 10 0 10 1 10 2 Huge phase lag! 0 Summary and exercises Nyquist plot (Matlab command: “nyquist.m”) Nyquist (Matlab Nyquist stability criterion for feedback stability Nyquist Examples for Nyquist stability criterion Examples Exercises: For (half of) Nyquist plot below, Exercises: count N for each case. • A=-1 (Ans. N=2) A=- (Ans. • B=-1 (Ans. N=0) B=- (Ans. -2 0 0 0 -6 0 0 0 -2 10 10 -1 10 0 10 1 10 • C=-1 (Ans. N=2) C=- (Ans. B D F -4 0 0 0 2 E C A The phase lag causes instability of the closed-loop closedsystem, and thus, the difficulty in control. 2008/09 MECH466 : Automatic Control • D=-1 (Ans. N=0) D=- (Ans. • E=-1 (Ans. N=2) E=- (Ans. • F=-1 (Ans. N=0) F=- (Ans. 29 2008/09 MECH466 : Automatic Control 30 Exercise Read Sections 10.3-10.5. Read 10.3Exercise: Suppose L(s) is stable and has Exercise: L(s) Nyquist plot below. Find the range of OL gain K>0 for which CL system is stable. -1 -3.3 -2 -0.5 (Ans. 0<K<1/3.3, 1/2<K<2) 2008/09 MECH466 : Automatic Control 31 8...
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