Nyquist criterion from given pole-zero and Nyquist plot

Nyquist stability criterion

For a negative feedback system with open-loop transfer function $G(s)H(s)$:

$$Z = P + N$$

• $P$ = number of open-loop poles in the right half s-plane;
• $N$ = net number of clockwise encirclements of the point $(-1, j0)$ by the Nyquist plot ($N$ is negative for counter-clockwise encirclements);
• $Z$ = number of closed-loop poles in the right half-plane.

The closed loop is stable if and only if $Z = 0$, i.e. the Nyquist plot must encircle $(-1, j0)$ counter-clockwise as many times as there are open-loop RHP poles. If P = 0, it must not encircle −1 at all.

Applying it to the given system

• From the pole-zero plot: open-loop poles at $s = -3$ and $s = +2$, and a zero at $s = -1$. So $G(s)H(s) = \dfrac{K(s+1)}{(s+3)(s-2)}$, with P = 1 (the pole at +2).
• From the Nyquist diagram: the plot crosses the real axis at about −1.25 and 0, so the point −1 lies inside the closed curve. The curve is traversed counter-clockwise, so it encircles −1 once anticlockwise: N = −1.

$$Z = P + N = 1 + (-1) = 0$$

The closed-loop system is stable.

(Consistency check: at ω = 0, $G(0) = -K/6 = -1.25$ gives $K = 7.5$. Then $1 + G = 0$ gives $s^2 + 8.5s + 1.5 = 0$, whose roots are both in the LHP.)