There is a question which I'm wondering again and again in recent months. I have taken courses on elementary differential equations, signals and systems, linear control systems, general theory of circuits and networks but I still do not know how can a simple linear integral transformation (Laplace transform) be so much helpful in analyzing and designing linear systems. For example, I think I somewhat know what "time domain" is, but how about the "frequency domain"?
Consider a block diagram in the frequency domain. Conceptually, what do the blocks represent? What is traveling(?) on the connections between blocks? First, I have thought that instead of taking care of the statement "when does some phenomenon happens" we consider "how many times does a phenomenon oscillate", but how can this approach be more helpful? Is it the best (optimal) way of treating such problems?
I think control systems and even systems have a more general meaning in mathematics. I even think that the very problem of "controlling a system" does have multiple "paradigms" — A.I., neural networks, state space approach, fuzzy logic, robust control, adaptive control, etc.
What I'm looking for is a book (preferred) or paper discussing the very beginning notions of system, control system, its history, and more importantly, what is the field or language in mathematics to study systems rigorously? Dynamical systems? Game theory? Differential equations? Linear algebra? A.I?
Any help is appreciated. Thanks.
P.S.: I've already studied these books:
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Alan Oppenheim's Signals and Systems (I think it is the best in the field)
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Ogata's Modern Control Engineering
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Desore's Basic Circuits Theory
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Charles A – Kuh and Ernest S's A rather brief book on the history of control systems ($H_\infty$ methods and so on, but I didn't get much; at least, this book induced some information on me that nobody actually knows what "feedback" is, why is it so helpful and why we should take samples of the output! It wasn't a precise book, I admit)
26 May 2013:
As I said, I am looking for a book or article discussing the origins of Control Theory and Systems. Seven days are passed since I started the bounty, yet I didn't find an answer. I'll get to the bottom of this issue as soon as I get some free time. Thanks.
Best Answer
Linear Algebra Underlies Everything.
The power of the Laplace transform derives from the power of concepts like a linear operator and an eigenfunction. The exponential is the eigenfunction of the derrivative operator, which is the main operator in control theory. By projecting the system onto bases which are the eigenfunctions of the operators in your system, you simplify the problem by exposing the symmetries. This is what the Laplace transform does ($\int f(x) e^{-sx}dx$ is like an inner product between co-ordinates ($f(x)$) and the new bases you want to represent your function/vector in (exponentials). The result are new co-ordinates in the exponential space)
The complex exponential is the eigenfunction of the second derrivative operator. So projections into this space expose a different set of symmetries, in this case, the 'frequencies'. So the Fourier Transform is also just linear algebra.
I'd recommend a healthy dose of linear algebra, to satisfy all your inquisitive needs!