- 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.
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!