What is the reason that when we minimize the dc gain of the asymptotically stable LTI system, the poles of the system is shifted in left-hand side or the transient response of the system is improved effectively?
you have considered only the real poles, but if you include the complex poles too, then the general formula doesnt hold.
DC gain can be lower and as the pole is shifted to the left the DC gain increases.
DC gain=T(s) overall transfer function at s=0 for a unit step input.
so as the closed loop poles are so far on the left HSP from the imaginary axis, the dc gain will be smaller and the transient response will be faster.
DC gain inversely with the CLP
Pravin, it seems the DC gain gets smaller as a CONSEQUENCE of the poles getting shifted towards the left half of the plane and not otherwise. Did u intend to mean it in any other way ?
Dear Pravin,
I think the best response for your specific question you get if you plot the Root-Locus of the open-loop system.
Root-Locus gives you the locations of the closed-loop poles as a function of the gain. You can see that, as the gain changes from 0 to infinite, the closed-loop poles location moves from the open-loop poles towards the open-loop zeroes.
(Any plant has the same number of poles and zeros, if we consider the zeros at infinite.)
So, not only your open-loop system must be stable, but also the open-loop poles must located well in the left half-plane for you to get "better" stability if you reduce the gain. Otherwise, you may get any behavior when you change the gain.
Actually, your opinion represents what used to be the general engineering opinion: if you increase the gain, you affect the stability, so better use low-gains.
It was Nyquist at AT&T, who reduced the gain to improve stability and was surprised to "discover" that the expected "improvement" actually was total instability.
This was the start of Nyquist Theorem and plot, then Bode, Nichols, Root-Locus, etc.
As today "plotting" is just a word in MATLAB, it is worth plotting them all for different Transfer Functions and see what they tell you.
Have Fun!
Itzhak
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