I am trying to understand the conditions under which the parameters of a linear differential delay system are identifiable. I am reading this 2002 paper by Belkoura and Orlov: Identifiability analysis of linear time-delay systems. The following is paraphrasing:
The system
$ {\dot{x} }\left( t\right) =\Sigma _{i=0}^{r}A_{i}x\left( t-\tau_{i}\right) +\sum ^{l}_{i=0}B_{i}u\left( t-\tau _{i}\right) $
can be represented over the ring of polynomials $R[\lambda]$ as:
$ {\dot{x} }\left( t\right) =A(\lambda)x(t) + B(\lambda)u(t)$
where $ \lambda=[\lambda_1,...,\lambda_k] $ and $k$ is the max. number of non-commensurable time delay units in the system.
My understanding from Belkoura and Orlov is that the above system is identifiable if it is "weakly controllable" and if the input $u(t)$ is sufficiently rich. The latter, I understand, can be achieved if the input function has at least $(k+p)$ non-commensurable discontinuities of jump size $\neq 0$, where $p$ is the dimension of the input function. But, how can I tell if a system is weakly controllable? Also, please correct any misunderstandings above if you spot any - this is quite outside my usual field.
Weak controllability simply consists of the satisfiability of the Kalman rank condition of the time-delay systems when the matrices of the system take values in some ring of operators associated with the delay operators.
In other words, the system is weakly controllable if
$$\mathrm{rank}\begin{bmatrix}B(\lambda) & A(\lambda)B(\lambda)& \ldots & & A(\lambda)^{n-1}B(\lambda)\end{bmatrix}=n$$
and where the rank is taken over $R[\lambda]$.
For more information, you may look at the papers