How Transfer Function Zeros Affect Transient Response - Quick Concepts in Control 2

Zeros and their pull
Transient response unfolds
Poles, coefficients.
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The effect of transfer function zeros on system transient response is investigated. Two transfer functions with the same poles but different zeros are shown to have different transient responses. The functions are generalized and analyzed in the time and frequency domain. It is shown mathematically and graphically how zeros can interact with poles and scale coefficients in the time domain frequency response terms. The derivation proceeds from frequency to time domain and back to frequency domain, offering multiple perspectives on the effect of zeros.
Errata:
4/24/2024 There is an error when partial fraction is applied to H1(s) (time 1:01), the term (1-a) is 1 a is not present in the decomposition. This error was not present in the code so the visualizations do not change. Thanks @1jymelgar

[@SaidasRanade]

This is great stuff presented with simplicity and clarity. Thanks.

[@LearnGandC]

Thanks for the great comment. Those two qualities are what I strive for.

[@a.h.t.erangadesilva8320]

This is a very lucid presentation
Thank you

[@LearnGandC]

I appreciate it!

[@jay89boy]

very clear and highquality content. Thx alot for making theese Videos!

[@LearnGandC]

Thanks, Jay! Glad you like them. More coming soon. For convenience, you can freely access all my content organized by subject at www.learngandc.com.

[@ooocoo7383]

Thanks for your video， but could you talk about the physical meaning of pole point and zero point with us？

[@LearnGandC]

The question of physical meaning of zeros would be a good future topic. Zeros occur at specific frequencies. From the vantage of a transfer function, this means that if you provide input to the system at this frequency with finite amplitude, there would be no output. This would be like providing a sinusoidal force to a system containing masses, springs, and dampers, and somehow the masses don't move. The zeros affect the transient response of the system, but do not affect the stability of the system like the poles. The poles dictate the characteristic response of the system states, like decay, oscillation, or amplification.

[@1jymelgar]

there is an error when partial fraction is applied to H1(s) (time 1:01), the term (1-a) is 1 a is not present in the decoposition.

[@LearnGandC]

Thanks for catching that. It's good to know at least one person is working through the material! In my code, I believe I had it correct so all results should reflect -a and not 1-a.

[@luisrebhan3268]

To be clear, the result of the partial fraction expansion should be:
1/(p*-p)/(s+p) + 1/(p-p*)/s+p*) right?