-3 dB Bandwidth Function Appears To Only Work For Specific Transfer Functions #1213

That nan is expected here, and it is not about series or tf, it is the integrator.

ct.bandwidth is defined as the first frequency where the magnitude drops dbdrop dB (default -3) below the DC gain, abs(H(0)). Your open-loop L = GsGc has a pole at the origin (Gs = 20/(0.5 s^2 + s) = 20/(s(0.5 s + 1)) is an integrator), so its DC gain is infinite. With no finite low-frequency level to measure the -3 dB point against, the bandwidth is undefined and the function returns nan. It is documented: the docstring says it returns "nan if the system has infinite dc gain", and the code just does if np.isinf(self.dcgain()): return np.nan.

So the pattern is "has an integrator or not", not "simple vs combined". Gc, sys1/2/3 and sys4 all have a finite DC gain, so they return a value. If you call ct.bandwidth(Gs) on its own you get nan too, because Gs has the integrator. I think that is the part that looked inconsistent.

What to use instead depends on what you want. If you want the loop bandwidth, take it on the closed loop, which has a finite DC gain:

T = ct.feedback(L, 1)
print(ct.bandwidth(T))     # about 2.59 rad/s here

If you want an open-loop number, "-3 dB below DC" does not apply to an integrator. The usual open-loop measure is the gain-crossover frequency, where abs(L(jw)) = 1 (0 dB):

gm, pm, wcg, wcp = ct.margin(L)
print(wcp)                 # gain-crossover frequency, about 1.58 rad/s here

Both give you a finite number for your L(s).