This question is too broad as it stands. There is a longer answer already, but I think it is possible to deal with a core part of the question easily.
The concrete question is what is a state-space representation, if possible with some of the intuition, its uses in economics (control theory)
My background is in control systems theory, where the notion of a state space representation developed. Optimal control theory, including the Kalman filter, was developed mainly in the 1960s. Economists have kept using the core ideas, but migrated off to their own standards of mathematical presentation.
A state space model is a canonical representation of a mathematical system. The idea is to take a model derived from underlying dynamics, and re-cast it into a standard format, so that existing tools can be applied to it.
In control theory, the standard canonical form of a nonlinear state space system is:
x(t+1) = f(t, x(t), w(t)),
y(t) = g(t, x(t), w(t)),
x(0) = x_0 \in R^n,
- $x(t)$ an n-dimensional state vector,
- $w(t)$ a vector of all external variables,
- $y(t)$ a vector of measured variables.
A fairly typical simplification is that the system is linear and time invariant, where the description collapses to:
x(t+1) = A x(t) + B w(t),
y(t) = C x(t) + D w(t).
Typical examples of external variables that appear in $w$ include:
- variables used to control the system (typically fixed by feedback rules),
- variables that disturb the state variables,
- noise that effects measurements.
The vector $w$ is often partitioned into groups corresponding to these cases, and the $B,C,D$ matrices are correspondingly partitioned.
Once we have the system in this format, we can then apply existing tools. A typical example is applying the Kalman filter to infer the state variable $x(t)$ from directly measured variables. Numerical toolboxes will give results based on the $[A,B,C,D]$ representation.
In economics, the variable $r^*$ is typically inferred by the use of a Kalman filter (E.g., see New York Fed r* page). This variable roughly corresponds to what was called “the natural rate of interest”, and cannot be measured directly. Instead, it needs to be inferred from measured variables. This example underlines why we need to differentiate between the state vector, and the variables that are directly measurable. The state vector includes all variables need for the dynamical description of the system, and not just variables that are measured.
(Note: control theory uses a somewhat unintuitive definition of “observable.” A variable is observable if its value can be inferred from measurements - even if it is not directly measured. Since observability is a mathematical requirement for the Kalman Filter to converge, this needs to kept in mind when reading documentation prepared by control engineers.)
As a final technical note, state space representations may not always exist. A key example in control theory are linear continuous time systems with a time delay. It would require an infinite number of states to represent a time delay, and so engineers are forced to either use frequency domain representations, or an approximation.