SSR - matrix model of a dynamical system. It has the following form:
$$ {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t) \\ \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t) $$
Definition: The pair (A, B) is controllable if, given a duration T>0 and two arbitrary points x_0, x_T ∈ R^n, there exists a piecewise continuous function t→u(t) from [0, T] to R^m, such that the integral curve x(t) generated by u with x(0) = x_0, satisfies x(T) = x_T. http://cas.ensmp.fr/~levine/Enseignement/3Controllability.pdf
In other words: System is controllable if for any initial conditions (of state variables x) there exist input u(t) which in a finite time moves state variables to any (freely chosen) state.
There is three ways to check controllability of linear system:
Kalman’s controllability matrix (of size n×nm):
$$ {\displaystyle R=[B \; AB\; ... \; A^{n-1}B]} $$
If rank of R equals n, then system is controllable.
https://en.wikipedia.org/wiki/Gramian_matrix TODO: describe method or point out that it is difficult.
Hautus matrix is given by formula:
$$ {\displaystyle H(\lambda)=[\lambda I\! - \! A \:\: B ]} $$