Talk:Control Systems/Poles and Zeros

Physical Significance of Poles and Zeros
With respect to this article on poles and zeros of the transfer function, I find that the notion of poles and zeros in case of a simple transfer function is explained in detail and I specifically liked the part which details the "meaning of a pole", where the exponential and the sinusoidal nature of the response are decoupled and explained clearly.

However, I feel that it is also relevant to highlight the physical significance of the poles and zeros of a transfer function in this article. Personally, I have understood the physical significance of a pole as I have mentioned below. However, I still haven't understood the physical significance of a zero.

Physical Significance of a Pole: We can relate the statement "An (open loop)system has a pole at ω(rad/s)", to an object which has a natural frequency of oscillation of ω(rad/s). Furthermore, from the fundamental concept of resonance in physics, this object with a natural frequency "ω", will attain resonance when excited by an input of frequency "ω". And we know that resonance corresponds to maximum energy transfer. From the perspective of an (open loop)system, a pole at "ω", mathematically means that the denominator of the transfer function will become zero and hence, the transfer function reaches infinity. For practical reasoning, we can consider that, infinity in mathematical terms corresponds to some large energy physically. Accordingly, when the (open loop)system is excited by an input which has a frequency "ω", then there will be a large amount of energy transfer from input to output, which we also call resonance, and hence, the transfer function takes a value of infinity mathematically.

Kindly let me know with your feedbacks about my above view of the physical significance of a pole. I am equally keen to learn and know about other physical interpretations of pole and specifically a zero.

Thank You.Rajeev singh

Mbharatheesha (discuss • contribs) 20:37, 25 February 2011 (UTC)


 * Singh, I totally agree with you. All these books are just copy-paste of mantras. As I complained elsewhere, they do not explain the essence of things. I see that you can convert a recursion $$y_{n+2} = a y_{n+1} + b y_n + x_n$$ into generating function
 * $$z^{-2}(Y(z) - y_1 z^{-1} - y_0) = a z^{-1}(Y(z) - y_0) + b Y(z) + X(z)$$
 * and derive output
 * $$Y(z) = {X(z)z^{-2} + (y_1 - a y_0) z^{-1} + y_0 \over 1 - az^{-1} - bz^{-2}} = {X(z) + (y_1 - a y_0) z + y_0 z^2 \over (z - p_1)(z-p_2)}.$$

The poles of characteristic polynomial clearly characterize the system -- they are eigenvalues of the system and, thus, allow to compute n-th state quickly, by diagonalization. It would also be interesting to know how are they related to z. The generating functions approach, which I have taken, tells that the series variable z is not important for analysis. Yet, something suggest that poles, when $$z=p_i$$, the response Y(z) will be infinite. What does that mean? What is z if my input signal is X(z) and output is Y(z), how can I make my input to equal z and p_n to obtain infinite response? What does it mean that signal is equal to eigenvalue? Why z is frequency? Why you cannot have more zeroes than poles, is it causality or stability issue? --Javalenok (discuss • contribs) 17:20, 23 September 2014 (UTC)

Clarification
The article page says "Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively." ........ I'm focusing on the word 'frequencies'. I'm thinking, for example, about the Laplace transfer function 1/(s+2). The Laplace root of the denominator is '-2', which is not a 'frequency'. So, to avoid confusion, some kind of explanation needs to be added to the article to explain the meaning of "are the frequencies for which....". KorgBoy (discuss • contribs) 20:01, 6 August 2018 (UTC)