D-contour

At \(a\): \(s = 0\)
At \(b\): \(s = j\infty\) (imaginary)
At \(c\): \(s = \infty\) (real)
At \(d\): \(s = -j\infty\) (imaginary)
From \(a–b\): \(s = j\omega\)
From \(b–c–d\): \(s = re^{j\theta},\ r=\infty,\ -\pi / 2 < \theta < \pi / 2\)
From \(d–a\): \(s = -j\omega\)

The polar plot is a plot of \(G(s)K(s)\) for \(s\) from \(a\) to \(b\).

The Nyquist plot is a plot of \(G(s)K(s)\) for \(s\) on the D-contour.

Nyquist Stability Criterion

The Nyquist Stability Criterion (NSC) states that if the open-loop system has \(P\) number of unstable poles, then for the closed-loop system to be stable, the image of \(G(s)K(s)\) must encircle the point \((−1 + j0)\) \(P\) times anticlockwise.

Procedure

Plot \(F(s) = G(s)K(s)\) for all values of \(s\) around the D-contour. The Nyquist plot will always be symmetric with respect to the real-axis.
Count the number of clockwise encirclements around the \((−1 + j0)\) point, \(N\).
Determine the number of unstable (RHP) poles of \(G(s)K(s)\), \(P\).
Calculate the number of unstable closed-loop poles, \(Z\): \(Z = N + P\).

Poles on \(j\omega\)-axis

For systems with integrators or poles on the imaginary axis, the D-contour has to be modified to exclude the pole on the contour. This is because at these poles, \(G(s)K(s) = \infty\).

From \(a–b\): \(s = j\omega\) except at pole locations, \(e–f\)
From \(b–c–d\): \(s = re^{j\theta},\ r=\infty,\ -\pi / 2 < \theta < \pi / 2\)
From \(d–a\): \(s = -j\omega\) except at pole locations, \(g–h\)
At \(i–a\): \(s = \epsilon e^{j\theta},\ \epsilon \ll 1,\ -\pi /2 < \theta < \pi / 2\)
At \(e–f\) and \(g–h\): \(s = \epsilon e^{j\theta}\pm j\omega_{1, 2}\)