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  1. Can a system be defined as a mapping from a set of mappings, called input signals, to another set of mappings, called output signals, where the two sets of mappings may or may not have the same domain and/or codomain? I cannot find such a definition stated formally somewhere, and it is just my impression from engineering books and courses on signal and system.
  2. When the output and input signals share the same domain as being some ordered set such as $\mathbb{R}$ or $\mathbb{Z}$ which can be interpreted as time, in engineering, a dynamic system is a system whose output signal value at an instant time is a function of not only present but also past values of the input signal.

    • In Wikipedia, however

      In the most general sense, a dynamical system is a tuple $(T, M, Φ)$ where $T$ is a monoid, written additively, $M$ is a set and $Φ$ is a function $$ \Phi: U \subset T \times M \to M $$ with $$ I(x) = \{ t \in T : (t,x) \in U \}\,$$ $$ \Phi(0,x) = x\,$$ $$ \Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, \text{for} \, t_1, t_2, t_1 + t_2 \in I(x)\, $$

      I wonder how the engineering definition and Wikipedia's general definition are consistent?

    • If I am correct, in engineering, a feedback system is a system that use past output values as part of input. I wonder if an engineering/general dynamic system is also a feedback system, and if a feedback system is also an engineering/general dynamic system?

Thanks and regards!

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Are you looking for a mathematical perspective (I say perspective because you can always choose your own definitions) or an engineering one? Correct me if I'm wrong, but I have not encountered the terms system (in the sense mentioned in question 1.) and feedback much (if at all) in pure math (unless you count information theory). On the other hand, the article you cite has been written with mathematicians in mind. –  sai Apr 29 '12 at 1:56
    
@sai: Both perspectives are welcome. Thanks! –  Tim Apr 29 '12 at 2:04
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1 Answer

I had meant this to be a comment, but it's too long. I suspect that the above definition from Wikipedia does not provide for systems with feedback (or feedforward - while causality might be important for flying to the moon using Kalman filters, it is not an issue when performing image processing on the pixels of an image or looking at historical time series data). For instance, I would imagine that for feedback relying on $k$ other time-samples you need $\Phi:T\times M^k\to M$ and more generally you might want $\Phi:T\times \mathcal{M}_T\to M$ where $\mathcal{M}_T$ is a space of functions (possibly with time-varying support) for the case of continuous feedback or adaptive feedback (when you might want to look at differing windows of samples).

With regards to Question 1., here is what a standard undergraduate text (Signals and Systems by Oppenheim et al.) has to say:

A system can be viewed as any process that results in the transformation of signals. Thus, a system has an input signal and an output signal which is related to the input through the system transformation.

From personal experience, folks in electrical engineering are usually concerned with specific instances of feedback (the actual topology of the network under consideration). One rare instance of a result that holds for a large class of systems where we have feedback is that for the problem of communication through discrete, memoryless channels, feedback does not increase the channel capacity (for background, see channel capacity).

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