I'm not really an academic, this is mostly something I read about in bits and pieces. It's my own observation that roughly everything that needs a formal model can be expressed that way, the formulation of those two functions can be pulled out with varying degrees of difficulty.
In controls/linear dynamical systems for example, the canonical forms express it directly [1]. Same goes for anything with a clean state-space representation, like Markov Processes.
The notion of "observation" could also be called "doing something useful with state." For a lot of simple state models, like finite automata, the observation is the identity function.
As for the algebra/category theory, I'm not aware of any formal works that get into it (but also I don't keep up with the literature at that level). It's mentioned by various people (because the work was classified) but Shannon was the first to use topological representations of controls. Mason published these as "signal flow graphs" where multiplication/addition/differentiation/integration operations are represented by the edges and vertices of a directed graph, which he used to derive his famous gain rule [2].
My own interest in the abstract algebra related to this was the fact that the edge of a SFG is a multiplication by a state variable and constant (g * s|z^k) and vertices are summations. SFGs themselves are algebraic structures inside what I understand to be an Abelian group (I'm self taught on this, so forgive terminology/notation).
In the pro-audio niche there's a transform on SFGs called the Topology Preserving Transform and has various derivations (Andy Simper, Will Pirkle, Vadim Zavalishin all have writings on it). It's a fairly manual approach, so about a year ago I wrote an algorithm to do it by reformulating Zavalishin's algebraic derivation [3] as a graph transformation. The interesting bit is that when you look at it that way, the TPT is a homomorphism between Abelian groups (the former in the continuous time, the latter discrete time). At least that's how I understand it.
Anyway, this is mostly my self study. I like the abstraction and it makes composing systems very simple. I'm using these approaches in a Rust crate with some combinators on state evolutions/observations like one would iterators, I find it makes things very composable and elegant [4].
In controls/linear dynamical systems for example, the canonical forms express it directly [1]. Same goes for anything with a clean state-space representation, like Markov Processes.
The notion of "observation" could also be called "doing something useful with state." For a lot of simple state models, like finite automata, the observation is the identity function.
As for the algebra/category theory, I'm not aware of any formal works that get into it (but also I don't keep up with the literature at that level). It's mentioned by various people (because the work was classified) but Shannon was the first to use topological representations of controls. Mason published these as "signal flow graphs" where multiplication/addition/differentiation/integration operations are represented by the edges and vertices of a directed graph, which he used to derive his famous gain rule [2].
My own interest in the abstract algebra related to this was the fact that the edge of a SFG is a multiplication by a state variable and constant (g * s|z^k) and vertices are summations. SFGs themselves are algebraic structures inside what I understand to be an Abelian group (I'm self taught on this, so forgive terminology/notation).
In the pro-audio niche there's a transform on SFGs called the Topology Preserving Transform and has various derivations (Andy Simper, Will Pirkle, Vadim Zavalishin all have writings on it). It's a fairly manual approach, so about a year ago I wrote an algorithm to do it by reformulating Zavalishin's algebraic derivation [3] as a graph transformation. The interesting bit is that when you look at it that way, the TPT is a homomorphism between Abelian groups (the former in the continuous time, the latter discrete time). At least that's how I understand it.
Anyway, this is mostly my self study. I like the abstraction and it makes composing systems very simple. I'm using these approaches in a Rust crate with some combinators on state evolutions/observations like one would iterators, I find it makes things very composable and elegant [4].
[1] https://www.engr.mun.ca/~millan/Eng6825/canonicals.pdf
[2] https://en.wikipedia.org/wiki/Mason%27s_gain_formula
[3] https://www.native-instruments.com/fileadmin/ni_media/downlo...
[4] https://github.com/m-hilgendorf/dsp64