Nyquist plot of (s+2)/[s(s+1)(s+3)]

$$G(s)H(s) = \frac{s+2}{s(s+1)(s+3)} = \frac{\tfrac23\left(1 + \tfrac{s}{2}\right)}{s(1+s)\left(1 + \tfrac{s}{3}\right)}$$

Open-loop poles: $0, -1, -3$, so P = 0. There is one pole at the origin, so the contour is indented there.

1. Magnitude and phase

$$|G(j\omega)| = \frac{\sqrt{\omega^2 + 4}}{\omega\sqrt{\omega^2 + 1}\sqrt{\omega^2 + 9}}, \qquad \angle G = -90^\circ + \tan^{-1}\frac{\omega}{2} - \tan^{-1}\omega - \tan^{-1}\frac{\omega}{3}$$

2. Real and imaginary parts

$$G(j\omega) = \frac{(2 + j\omega)\left[-4\omega^2 - j(3\omega - \omega^3)\right]}{16\omega^4 + (3\omega - \omega^3)^2}$$ $$\text{Re} = \frac{-5\omega^2 - \omega^4}{16\omega^4 + (3\omega - \omega^3)^2}, \qquad \text{Im} = \frac{-6\omega - 2\omega^3}{16\omega^4 + (3\omega - \omega^3)^2}$$

• Im < 0 for every ω > 0, so the plot never crosses the real axis at a finite frequency. The phase approaches −180° only as ω → ∞.
• Low-frequency asymptote: as ω → 0, Re → $-5/9 = -0.556$. (Equivalently $K\sum T = \tfrac23(1 + \tfrac13 - \tfrac12) = 0.556$.)

3. Table (ω > 0)

4. Mapping of the full contour

• ω = 0⁺ → +∞: from $\infty\angle-90^\circ$ (along Re = −0.556) down to the origin, arriving at −180° (third quadrant).
• ω = −∞ → 0⁻: mirror image in the second quadrant.
• Infinite semicircle: maps to the origin.
• Small indentation at s = 0: $G \approx \tfrac23/s$ maps to an infinite arc, clockwise through the right half-plane (from +90° to −90°).

                       ..              |
                        .              |
                        ..             |
                         ..            |
                          ..           |
                           ...         |
                             ...       |
                               .....   |
-------@---------------------------*****----------
                               *****   |
                             ***       |
                           ***         |
                          **           |
                         **            |
                        **             |
                        *              |
                       **              |
                  -0.556               0
* w > 0   . w < 0 (mirror)   @ point (-1, j0)
asymptote Re = -0.556; no negative real crossing

5. Stability

Nyquist criterion: $Z = P + N$, where $P$ = number of open-loop poles in the RHP, $N$ = net clockwise encirclements of $(-1, j0)$ by the Nyquist plot of $G(s)H(s)$, and $Z$ = number of closed-loop poles in the RHP. The closed loop is stable only if $Z = 0$.

The plot never cuts the negative real axis, so $(-1, j0)$ is not encircled: N = 0, and $Z = 0 + 0 = 0$.

The closed-loop system is stable. Also, GM = ∞ and PM = 64.8° (at $\omega_{gc}$ = 0.588 rad/s).

Check: $s^3 + 4s^2 + 4s + 2 = 0$ gives Routh first column $1, 4, 3.5, 2$, all positive. ✓