Bode plot of 1000/[s(1+0.1s)(1+0.001s)]

Factors and corner frequencies

$$G(s)=\frac{1000}{s(1+0.1s)(1+0.001s)}$$

• $K=1000$ (60 dB), one pole at origin: initial slope $-20$ dB/dec.
• Corner frequencies: $\omega_1=1/0.1=10$ rad/s, $\omega_2=1/0.001=1000$ rad/s.

Asymptotic magnitude plot

$$\begin{aligned} \omega=1:&\ 20\log1000 = 60\ \text{dB}\\ \omega=10:&\ 60-20 = 40\ \text{dB}\\ \omega=100:&\ 40-40 = 0\ \text{dB}\\ \omega=1000:&\ 0-40 = -40\ \text{dB}\\ \omega=10^4:&\ -40-60 = -100\ \text{dB} \end{aligned}$$

Phase plot

$$\phi(\omega)=-90^\circ-\tan^{-1}0.1\omega-\tan^{-1}0.001\omega$$

 dB 60 |\  -20
    40 |  \___ w=10
     0 |-------\---------------- w
       |   w=100\  -40
   -40 |         \__ w=1000
       |            \ -60
 deg   |
  -90  |--__
 -180  |-------x---------------- 
       |      100   \___
 -270  |                 ----
       1    10   100  1000  1e4

Crossover frequencies and margins

• Phase crossover: $\tan^{-1}0.1\omega+\tan^{-1}0.001\omega=90^\circ \Rightarrow 0.1\times0.001\,\omega^2=1 \Rightarrow \omega_{pc}=100$ rad/s.

$$|G(j100)|=\frac{1000}{100\sqrt{1+100}\sqrt{1+0.01}}=0.990,\quad GM=-20\log0.990=+0.086\ \text{dB}$$

• Gain crossover: the asymptote gives 0 dB at $\omega=100$ rad/s; exactly $\omega_{gc}=99.5$ rad/s, where $\phi=-179.94^\circ$.

$$PM=180^\circ-179.94^\circ\approx0.06^\circ$$

Comment on stability (1 mark)

$\omega_{gc}$ and $\omega_{pc}$ almost coincide, and GM ≈ 0 dB, PM ≈ 0°. The closed-loop system is just stable (on the verge of instability, practically marginally stable): its dominant poles are at $-0.049\pm j99.5$, giving a very lightly damped oscillation at about 100 rad/s. Any small increase in gain makes it unstable. Routh check: $0.0001s^3+0.101s^2+s+1000$: $0.101\times1=0.101>0.0001\times1000=0.1$, so stable by a very small margin. A compensator (e.g. lead) is needed for acceptable relative stability.

Answer: $\omega_{gc}\approx\omega_{pc}\approx100$ rad/s, GM ≈ 0.09 dB, PM ≈ 0.06°: the system is barely (marginally) stable.