Monday, September 17, 2007

perfect and the dirac function

A question i was asked you believe in perfect... it was an odd question to be asked. i probably wouldn't have pondered so much about it if it had been "perfection" and gone on a fairly long monologue on my idea of perfection. but 'perfect', had a sense of ambiguity and enigma surrounding it.
as i thought about it myself, i went back to our dynamic modeling class earlier today. it was a regular class like any other. concepts we learned in 12th std being refreshed, about the dirac-delta function, and how every function could be expressed as an infinite summation of weighted impulse functions convoluted over time. but somehow today, i was seeing things in a different light. i dunno now if it was Shorya's way of communicating it that made me see things in a new light, but the convolution integral had always been a conundrum for me. its one of those concepts you always understand when you are with it, but fades away, becoming hazy. somewhere in the beauty of all that mathematics that Dirac, Euler and Laplace defined for us, was the answer to my friend's question. I went back to the class time and again today, as i idly passed away the time in the lab, thought about wonderful way in which every mathematical aspect fell into place. starting from the definition of the dirac function, our consideration of an ideal impulse function, weighted with the function being mapped, its correlation with linearity and super position, and then finally the laplace integral. As we went deeper into the mathematics, and made our assumptions that the system considered was time invariant and linearised, i realized that so much for the mathematics, yet no one equation could be 'perfectly' used to define a real time system. despite all that, we still find a way around it, make a few assumptions, and viola, we had the foundations of the classical control theory.
There is no human relationship that is perfect, yet treated with slight adjustments, there is always a way to work around such imperfections and control it so that it reaches a comfortable equilibrium. Life itself could be treated as an infinite summation of impulse functions or events. some weighted high, like the moments of extreme happiness you get when you receive a letter from a loved one. intense spikes, that instantly attenuate. The effect of such an event or pulse is however felt later on. In a way, experience is the sum total of our response to such events collected from the finite time of our birth. the effects of which might be felt a long time after the actual event. the insignificant events damping out sooner, their poles being so much farther away in the left half of the s-plane. events that get under our skin.. irritating, marginally destabilizing. but not threatening. you can chose to ignore it, and get it under control. there are these intense moments of sadness as well, undamped and unconquerable,taking us away to the right half leaving us unstable, waiting for an external influence, a buffer, a support to prop us back into reality. yet no matter what control algorithm you use, they still leave behind a permanent scar, a steady state error. that remains with you. there will never be anything called a free lunch.always a trade-off. a compromise.. we as individuals will never be time invariant, yet, we are also adaptive, always changing. learning from past experiences. sometimes learning quickly. other times refusing to learn. letting the same repeated events bother us. letting others become routine.
getting back to the question, none of the classical control theory which we study would have been possible if its weren't for certain assumptions made. It all starts with an impulse. Somewhere back in time. Life itself is like that. there can never be a perfect spread of time. but if we are able to freeze every single point in time. and record our response to that instantaneous event. we would see there is indeed that elusive perfect.
PS: i think i will make this a floating blog. My own thoughts on this matter are a little incoherent as of now. should improve with time. hopefully.


Puthali said... a bell in some remote corner of ma mind... :D
neways nice post...btw y do u say...there's nothing like free-lunch? if by tht, do u mean tht there's always a motive? or in case of no u consider self-satisfaction one gets, itself as a motive?
btw was just listening to song called 'free love' by Depeche mode...nice one...wonder wht u will have to say to tht... ;)

Akshay said...

Pantramma perfection function yella pursue maadtheera!!!