NYQUIST CRITERION
The Nyquist criterion is one of the widely used stability analysis techniques in the s-plane, based on the frequency response of the system. To determine the frequency response of a continuous system transfer function G(s), we replace s by j ω and use the transfer function G( j ω). In the s-plane, the Nyquist criterion is based on the plot of the magnitude |GH ( j ω)| against the angle I GH ( j ω) as ω is varied.
In a similar manner, the frequency response of a transfer function G(z) in the z-plane can be obtained by making the substitution z = e j ωT . The Nyquist plot in the z-plane can then be obtained by plotting the magnitude of |GH (z)|z=e j ωT against the angle I GH (z)|z=e j ωT as ω is varied. The criterion is then
where N is the number of clockwise circles around the point −1, P the number of poles of GH(z) that are outside the unit circle, and Z the number of zeros of GH(z) that are outside the unit circle.
Fora stable system, Z must be equal to zero, and hence the number of anticlockwise circles around the point –1 must be equal to the number of poles of GH(z).
If GH(z)has no poles outside the unit circle then the criterion becomes simple and for stability the Nyquist plot must not encircle the point −1.
An example is given below.
