If I want to do a bode plot of a transfer function who looks like this : $$ G (s) = \frac {Ke^{-Ts}}{s^2 + 2s + 3} $$
Should I just compute the bode plot by replacing $s $ to $j\omega $ and then find the amplitude and phase. When it comes to delay, only the phase will be effected.
Or should I replace $e^{-Ts} $ with Padé approximation?
Or should I transform the transfer function to discrete transfer function?
Best Answer
You can use a Padé approximation or discretize the system when dealing with delay. However both these methods will always be an approximation of the actual transfer function, so at sufficiently high frequencies they will always start to diverge significantly. But if you choose a Padé approximation of high enough order or discretize the system with a small enough sample time then the frequencies at which this happens can get arbitrarily high.
The advantage of Padé is that the magnitude does not get distorted, however the phase does. Discretizing a system distorts both, but the phase distortion is typically less, however it is only really defined up until the Nyquist frequency. For comparison take a look at the figure below, which plots the Bode diagram of your given transfer function with $K=1$ and $T=1$.
If you want an exact representation of your transfer function then the only option would be to evaluate it manually. So calculate the magnitude and phase for the desired frequencies ($s=j\,\omega$). The only downside to this is phase wrapping (from a single complex number the phase will always be in a range of 360° or 2$\pi$). You can try to correct this by detecting which multiple of a full rotation would yields the smallest phase jump compared to the previous phase. However this has the downside that this does not work when you have poles and zeros very close to the imaginary axis, for example
$$ G(s) = \frac{1}{(s^2 + 1)^2}, $$
has a jump of -360° in its actual phase. So an advantage of the previous two methods would be that you probably still can use the build-in function of Octave to plot the Bode diagram and not have to deal with phase wrapping, however I am not sure if they would be able to handle jumps of 360°.