S+T=1 sensitivity of a control system

[FOIWATER]

with s being the sensitivity of a transfer function GC/(1+GC) to the parameter G. (1/(1+GC)

I see mathematically how it works out. My question is about the profoundness of the statement. Qualitatively, what does it mean

 

[jim hardy]

My question is about the profoundness of the statement. Qualitatively, what does it mean

I gather you're talking about feedback systems here...

To me it is a demonstration of the profound effect of feedback...

G is the forward gain

and C is the feedback
let C be +1 for simplicity's sake

observe that gain of the feedback loop G/(1 + G) cannot be greater than 1, regardless of forward gain G,,,,

unless you reverse sign of feedback
which creates possibility of a zero denominator (hence infinite closed loop gain) hence oscillation...

And that happens by itself when time lag of feedback amounts to a half cycle(180 degrees) which reverses the sign...
Any physical system with feedback will find that frequency for you; recall from elementary school when the PA microphone gets in front of the speaker you get a pure tone - that's the frequency where transit time at speed of sound from speaker to mike amounts to a half cycle.

Mother Nature is very playful !
Look at Earth's feedback system for temperature control - heat transport from surface to stratosphere via water vapor.

old jim