What is the mathematical foundation of Control Theory?

Question

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.

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P.S.: I've already studied these books:

• Alan Oppenheim's Signals and Systems (I think it is the best in the field)

• Ogata's Modern Control Engineering

• Desore's Basic Circuits Theory

• 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)

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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.

Answer 1

Arbib and Manes once made ​​an attempt to apply the Automata Theory to Control Systems, see:

Manes, E.G.(ed.) Category theory applied to computation and control. Proc. $1$st Internat. Symp., San Francisco, 1974. Lecture Notes in Comp. Sci., 25.

I am not an expert in the Control Systems and can not judge how successful this attempt is, but from the point of view of the Automata Theory, it looks pretty well.

Answer 2

All mathematics, functions, transforms and ..., can just model a physical system and sometimes all these models dont work properly. The blocks in linear control systems, are just models to let you understand better about the physics, say voltages, currents or maybe fields, that travel(!) from one par of the system to another… The reality of a control system is the physics not the blocks in frequency domain, but these block just try to let you understand it better and let you control it when its going to be unstable. Think of Fourier series which is indeed, expansion of a function to an infinite sum of complex exponentials, sometimes we dont need all of these exponential to model a physical waveform and a prat of them work, and sometimes the whole series fail to model a specific waveform.

A little bit Electronics (active circuts with feedback), Linear algebra and measure theory help you to understand better about the physics and foundation of Control theory...

Answer 3

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!

Answer 4

Control theory studies systems that have an internal state and can be connected with other systems via inputs and outputs. The dynamics are often but not always described by differential or difference equations.

Because of its engineering motivations, control theory is often interested in detailed analysis results and design techniques. Therefore it tends to spend more time on the easier cases: linear-time invariant systems. Linear algebra, and Laplace and Fourier transforms are extremely useful techniques, and about the most central ones.

Other relevant areas of math include differential geometry, dynamical systems, probability, algebra, analysis, automata, complexity of algorithms, you name it. Control theorists and engineers are always looking to apply other mathematical techniques to their problems, so no one of them is control theory.

As for references,

Feedback Systems: An Introduction for Scientists and Engineers Karl Johan Åström & Richard M. Murray http://press.princeton.edu/titles/8701.html

is a very good book whose Chapter 1 begins with sections "1.1 What Is Feedback? 1.2 What Is Control?" Do not expect precise mathematical definitions but do consider consigning the text that states "nobody knows what feedback is" to the recycling bin.

A more mathematical and more advanced but equally excellent text is Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition, Springer, New York, 1998 http://www.math.rutgers.edu/~sontag/mct.html