M6.3 Bode Analysis

Here, we enter the first-order + time-delay model in the MATLAB command window.

» fodt = tf([1],[10 1],'InputDelay',5)

Transfer function:
   1
--------
10 s + 1

Input delay: 5

Now, let us perform a Bode analysis assuming a proportional controller gain of 1.

» kc = 1;
» [mag,phase,w] = bode(kc*fodt);
» subplot(2,1,1), loglog(w,squeeze(mag))
» subplot(2,1,2), semilogx(w,squeeze(phase),[0.01 1],
  [-180 -180],'--')

The resulting Bode plot, with the –180° line drawn for convenience, is shown in Figure M6-3. Notice that the crossover frequency is around 0.36 rad/min, where the amplitude ratio is approximately 0.26. The gain margin is then 1/0.26 = 3.8. This means that the controller gain could be increased to roughly 3.8 before the process goes unstable. This is consistent with the finding of k_c = 3.8 for instability, shown in the previous section (time-domain simulation).

The gain and phase margins can be calculated using the following MATLAB function:

» [Gm,Pm,Wco,Wpm] = imargin(squeeze(mag),squeeze(phase),w)

Gm = 3.80093442825351
Pm = Inf
Wco = 0.36655875121677
Wpm = NaN

where w_co is the crossover frequency (where f = –180°) and w_pm is the frequency where AR = 1.

Gain Margin

The gain margin of 3.8 indicates that the controller gain can be increased from 1 to 3.8 before the process becomes unstable—again, this is consistent with the previous results. Also notice that the phase margin is infinity. This is because the amplitude ratio is always less than 1 (it is exactly 1 only at 0 frequency, when the proportional gain is 1).

Crossover Frequency/Ultimate Period

The crossover frequency is 0.366 rad/minute. Converting this to cycles/minute, we find a frequency of 0.0583 cycles/minute, which corresponds to a period of 1/0.0583 = 17.16 minutes. This is exactly the value found in the time-domain simulations.

Nyquist Diagram

The Nyquist plot (Figure M6-4) can be generated by using the following commands:

» [re,im,w] = nyquist(kc*fodt);
» plot(squeeze(re),squeeze(im))

Problem 1. Use a controller proportional gain of 3.8 and perform a Bode analysis to show that the amplitude ratio is 1 at the crossover frequency. What is the phase margin and frequency when AR = 1 (for the controller proportional gain of 3.8)?

Solution

The Bode and Nyquist plots for g_c(s)g_p(s) with k_c = 3.8 are shown in Figures M6-5 and M6-6. Notice that the phase margin is 1 (the AR = 1 at the crossover frequency) and the phase margin is 0 (the phase angle is -180° when the AR = 1).

This is also shown by the gain and phase margin analysis from MATLAB:

» kc = 3.8;
» [mag,phase,w] = bode(kc*fodt);
» [Gm,Pm,Wco,Wpm] = imargin(squeeze(mag),squeeze(phase),w)

Gm  = 1.00024590217198
Pm  = 0.03106191271067
Wco = 0.36655875121677
Wpm = 0.36646150971664

Notice that the gain margin is effectively 1, the phase margin is effectively 0, and the crossover and phase margin frequencies converge to 0.366 rad/min.