bandwidthcontrol systemsignal
This link says that the bandwidth is the frequency at which the magnitude of the frequency response is decreased by 3dB from the value of the initial horizontal asymptote.
But wikipedia says "In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak."
So, should the bandwidth be calculated from the initial horizontal slope or from the peak value ?
What you have initially described is a 2nd order low pass filter then you've made a bit of a complication of things because the bandwidth is \$\omega_n\$ i.e. \$\omega_n\$ is the 3dB point when \$\zeta\$ is not causing the filter to peak i.e. has a value of \$\frac{1}{\sqrt2}\$: -
OK you may be trying to derive a general expression but that doesn't help much visualize the problem so for now I'm assuming \$\zeta\$ = \$\frac{1}{\sqrt2}\$.
Anyway, moving on and ignoring your expression for \$\omega_0\$, the bandwidth of the low-pass filter is from dc to \$\omega_n\$. Then you've modified the 2nd order low-pass filter expression with a 1st order high pass filter (\$s +z\$).
When s is very low, the low pass filter is unaffected other than having a "gain factor" of z. If z is unity then the low pass filter response is unaffected until jw approaches z in magnitude - this then marks a lower 3dB point and the net response climbs with increasing frequency until the original \$\omega_n\$ is reached then, because the low pass filter is a second order, the response starts to fall again.
What you have proposed is a band pass filter with finite gain at dc.
I'm not sure about the bandwidth
If the high pass filter (\$s +z\$) comes into play at significantly lower frequencies than \$\omega_n\$ then the net bandwidth reduces from \$\omega_n\$ to \$\omega_n - z\$.
That's how I see it anyway.
If you want a amp with low distortion, look at the distortion numbers. It shouldn't matter to you how that is achieved under the hood, only that it is.
There are all kinds of ways to trade off parameters in a audio amplifier. Taking one or two parameters in isolation doesn't mean anything. Look at the result, not the method.
That said, simply using large open loop gain, then global feedback to fix everything, isn't how it's generally done. The problem is that such a system tends to suffer from TIM (transient intermodulation distortion). The better systems tend to use some local feedback in each stage, with the overall open loop gain not wildly above the desired closed loop gain. Then moderate application of negative feedback keeps the closed loop gain predictable and the frequency response flat.
Again though, measure results, not how they were achieved. There is more than one way to design good audio amplifiers.
Best Answer
It seems that you are speaking about the bandwidth of a lowpass system, right?
At first, you should realize that the bandwidth of a system is a parameter that needs DEFINITION. Hence, in some cases, you can define the bandwidth for your own purposes even at 1dB or 2dB points. But that it is not the core element of your question.
(1) Your graph shows a lowpass response according to a so called "Chebyshev behaviuor" (with gain peaking higher than the value at f=0). Indeed, for such a transfer function it makes sense to specify the bandwidth at that frequency where the magnitude crosses again the DC value (horizontal line).
(2) However, if the magnitude response does NOT exhibit such a peaking (lowpass response according to a "Butterworth" or "Bessel") it is commonly agreed to use the classical 3dB frequency (3 dB below the DC value).
Summary: The bandwidth of a lowpass system is a matter of definition. In general, it defines the end of the "passband" based on an acceptable "ripple" (magnitude variations) within this frequency band. If, for example, the gain peaking (in your figure) is w=1 dB it makes sense to specify that the ripple within the passband is 1 dB only. For another system without such peaking (typical example: Butterworth response) the magnitude variation (ripple) would be 3 dB.
Remarks: As you have made reference to a wikipedia contribution: Please note that the mentioning of "3 dB below peak" applies to a bandpass system only. And even this is a matter of definition - for some bandpass systerms it makes sense to require 1dB or 2dB break points. For lowpass responses the explanations/definitions apply as outlined above.
Please note that most filter design aids (tables, programs) are using bandwidth definitions in accordance with the above mentioned two cases.