EE128 Feedback Control Lecture 17, 10/24/2006

• Examples of Nyquist plots • Stability Margins (6.4) 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 curves (i.e. Bode plot) you can determine the stability of the closed-loop system without needing to determine the closed-loop poles. Y (s) KG(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) •N=0 Æ system stable unless there are open-loop poles in RHP (i.e. P≠0) •N<0 Æ system stable if P=-N, i.e. (one or more counterclockwise loops of -1/K) •N>0 Æ system unstable The argument principle

α θ r r jα H1(s0 ) =θv = v e φ = 1 + 2 − ( 1 +φ2 )

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 open- loop KG(s) determines stability of the closed loop system a(s) + Kb(s) 1+ KG(s) = a(s) N = Z − P

Y (s) KG(s) N = number of clockwise encirclements = T (s) = R(s) 1+ KG(s) Z = Zeros (closed-loop system roots) P = open-loop Poles in the RHP 1+ KG(s) = 0 Procedure for plotting Nyquist Plot Problem 6.18 Nyquist plot for a 2nd order system

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To determine range of gains K for which the system is stable Æ 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 Æ -1/K outside the loops Æ N=0 Æ all roots stable

The system is stable if |KG(jω)|<1 when the phase of G(jω) 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 Æ no freq response can be determined experimentally - But we can compute magnitude and phase and apply Nyquist

1 2 3 Stability Margins (6.4)

• Gain margin (GM): Factor by which the gain can be raised before instability results • Phase margin (PM): Amount by which the phase of G(jω) exceeds -180° when |KG(jω)|=1 • GM and PM are measures of how close the Nyquist plot comes to encircling the -1 point Stability Margins (6.4)

• Crossover frequency (ωc): frequency at which the gain is unity, or 0db. • Vector margin: distance to the -1 point from the closest approach of the Nyquist plot • Conditionally stable systems: systems that can be made stable by increasing the gain Stability Margins (6.4) Stability properties for a conditionally stable system Nyquist plot for system with multiple crossover frequencies