EE128 Feedback Control Lecture 16 10 19 2006 Outline The Frequency Response Design Method ch 6 The Nyquist stability criterion 6 3 The argument principle 6 3 1 Application to control design 6 3 2 The Nyquist stability criterion 6 3 Arose as a consequence of counterintuitive effects of feedback at Bell labs i e decreasing the gain would make an amplifier unstable Based on the argument principle it relates the open loop frequency response to the number of closed loop poles of the system in the RHP Useful for stability analysis of complex systems with more than 1 resonance i e when the magnitude crosses 1 or the phase 180 several times e g open loop unstable systems nonminimum phase systems etc The Nyquist stability criterion 6 3 The Nyquist stability criterion relates the open loop frequency response of KG s to the amount of roots of the characteristic equation 1 KG s i e zeros and poles that are in the RHP Advantage from the open loop frequency response e g Bode plot you can determine the stability of the closed loop system without needing to determine the closed loop poles KG s Y s T s R s 1 KG s a s Kb s 1 KG s 0 a s N Z P N number of clockwise encirclements Z Zeros of 1 KG s i e closed loop poles system roots in the RHP P Poles of KG s i e open loop poles in the RHP The Nyquist stability criterion 6 3 An encirclement of 1 by KG s is equivalent to an encirclement of 1 K by G s N 0 system stable unless there are open loop poles in RHP i e P 0 N 0 system stable if P N i e one or more counterclockwise loops of 1 K N 0 system unstable The argument principle r r j H 1 s0 v v e 1 2 1 2 Argument principle A contour map of a complex function will encircle the origin Z P times of the function inside the contour Application to control design Contour evaluation of an openloop KG s determines stability of the closed loop system 1 KG s a s Kb s a s N Z P Y s KG s T s R s 1 KG s 1 KG s 0 N number of clockwise encirclements Z Zeros closed loop system roots P open loop Poles in the RHP Procedure for plotting Nyquist Plot Problem 6 18 Nyquist plot for a 2nd order system 3 1 2 4 To determine range of gains K for which the system is stable consider G s only and count Number of encirclements of the 1 K point Nyquist plot for a 3rd order system Because of pole at origin For small Ks 1 K outside the loops N 0 all roots stable The system is stable if KG j 1 when the phase of G j is 180 Nyquist plot for an open loop unstable system Because of the pole in the RHP the system will never reach a steady state sinusoidal response for a sinusoidal input no freq response can be determined experimentally But we can compute magnitude and phase and apply Nyquist 1 2 3