Bode plot of 512(s+3)/[s(s²+16s+256)]

Time-constant form

$$G(s) = \frac{512(s+3)}{s(s^2 + 16s + 256)} = \frac{6\left(1 + \dfrac{s}{3}\right)}{s\left(1 + \dfrac{s}{16} + \dfrac{s^2}{256}\right)}$$

Quadratic factor: $\omega_n = 16$ rad/s and $2\zeta\omega_n = 16$, so $\zeta = 0.5$.

Factors and corner frequencies

Magnitude asymptotes

With ζ = 0.5 the resonant correction at $\omega_n$ is $-20\log(2\zeta) = 0$ dB, so the asymptotes are close to the true curve.

Phase

$$\phi(\omega) = -90^\circ + \tan^{-1}\frac{\omega}{3} - \tan^{-1}\frac{\omega/16}{1 - (\omega/16)^2}$$

Bode plot

   40 |*
      |*******
      |      ******
   10 |            **************
      |-------------------------****-----------------
      |                            ****
  -20 |                               ***
      |                                  ***
      |                                    ****
  -50 |                                       ****
      |                                          ****
      |                                             *
  -80 |
      +----------------------------------------------
       0.1        1          10          100        1000
Magnitude asymptotes (dB) vs w (rad/s)

    0 |
      |
      |
  -45 |                  ****
      |              *****  ***
      |        *******        **
  -90 |*********               **
      |                         *
      |                         **
 -135 |                          **
      |                           ***
      |                             ******
 -180 |----------------------------------************
      +----------------------------------------------
       0.1        1          10          100        1000
Phase (deg); -180 line shown

Margins and stability

• Gain crossover: the exact $|G| = 1$ at $\omega_{gc} = 24.4$ rad/s (asymptotic estimate 22.6 rad/s). There, φ = −137.9°, so

$$PM = 180^\circ - 137.9^\circ = 42.1^\circ$$

• Phase crossover: the phase tends to −180° only as ω → ∞, so it never actually crosses −180°. Hence $\omega_{pc} = \infty$ and GM = ∞.

Both margins are positive, so the closed-loop system is stable, with PM ≈ 42° (reasonably damped). Check: $s^3 + 16s^2 + 768s + 1536$ gives $16 \times 768 > 1536$. ✓

Answer: GM = ∞, PM ≈ 42.1° (at 24.4 rad/s). The system is stable.