EE 4314- Nyquist Plots

Updated: Monday, October 23, 2017

Making Nyquist Plots The Nyquist plot is a plot in the complex plane of Re(H(s)) and IM(H(s)) for sj  as  goes from zero to infinity. As such it captures in a single plot the two Bode plots of magnitude and phase versus  . The Nyquist plot is easily drawn from the Bode magnitude and phase plots.

Nyquist Plots of Compensators

Nyquist plots of the various compensators are important to understand since they give additional design insight. Here are representative Bode magnitude and phase plots, along with the corresponding Nyquist plot .

Bode plot for PD compensator

k p kK(s)  kd s  k p  kd (s  ) . kd kK() s ( s 1)

Nyquist plot

1 Bode plot for practical PD compensator k s  p k s 1 kK() s k d 10 . d ss 10 Note that this matches the previous plot for low frequencies. This is a lead compensator.

Nyquist Plot

Bode plot for PI compensator k s  i k s 1 kK() s k p . p ss

Note that phase is negative.

Nyquist plot

Bode plot for PID compensator k k s 2  p s  i k k kK(s)  k d d d s If the poles are real this can be written 1/T 1/T K 1 I D kK() s ( TD s 1) s sTI TT1, 1 / 1 dd Tii10, 1/ T 0.1 (ss 1)( 0.1) s2  1.1 s .1 kK() s  ss Note that the Bode plot looks like PI on the left and PD on the right.

Nyquist plot

Bode plot for lead compensator s  a kK(s)  k , a  b s  b s 1 kK() s  s 10 Note that phase is POSITIVE. That is why it is called ‘(phase) lead’.

Nyquist plot

Bode plot for lag compensator s  a kK(s)  k , a  b s  b s 10 kK() s  s 1 Note that phase is NEGATIVE. That is why it is called ‘(phase) lag’.

Nyquist plot

4 Bode plot for Complex Pole pair 2 n 25 kK() s 222 ss2225n ss 

Nyquist

5 Nyquist Stability Criterion

Harry Nyquist at Bell Labs in 1932

A basic feedback control system is the TRACKING CONTROLLER given in the figure. The plant is H(s) and the compensator K(s); the feedback gain is k. d(t) r(t)e(t) y(t) kK(s) H(s)

The closed-loop transfer function is Y (s) kK(s)H (s) T(s)   . R(s) 1 kK(s)H (s) The closed-loop poles are given as the root of the denominator (s) 1 kK(s)H (s) 1 kG(s) where G(s) is the open-loop gain. That is, the closed-loop roots are the solutions of 1()0kG s This requires two things Gs() 1/ k angle( kG ( s )) 180o This is used to develop the Nyquist stability criterion.

1. Determine the number of unstable poles of the open-loop gain Gs ( ) . Call this number P. 2. Make Nyquist plot of Gs(). Find the number of clockwise encirclements of the point  1 . k Call this number N. 3. The number of unstable closed-loop poles is Z=N+P

Note that if P is nonzero, then the Nyquist plot must encircle the point  1 P times in a k counterclockwise direction (e.g. N is negative) for the closed-loop system to be stable.

6 Gain Margin and Phase Margin

The gain margin and phase margin are defined in the Nyquist plot as shown