*3.1. Information Sufficiency*

Consider that the sampled external variables *w* are aligned in a vector of measurement data {*w*(1), *w*(2), ..., *w*(*T*)} of length *T*. Associated with this set of measurements, a *Hankel matrix* with a depth of *<sup>L</sup>* <sup>&</sup>lt; *<sup>T</sup>* <sup>∈</sup> <sup>Z</sup><sup>+</sup> can be defined as follows:

$$\mathcal{H}\_{L}(w) := \begin{bmatrix} w(t) & \sigma w(t) & \cdots & \sigma^{(T-L+1)} w(t) \\ \sigma w(t) & \sigma^2 w(t) & \cdots & \sigma^{(T-L+2)} w(t) \\ \vdots & \vdots & \cdots & \vdots \\ \sigma^L w(t) & \sigma^{(L+1)} w(t) & \cdots & \sigma^T w(t) \end{bmatrix} \tag{7}$$

Next, the persistency of excitation concept [26] is appealed to; to verify if the available information provided by measurements is sufficient to recuperate the system physical laws, we use. This is a condition that applies for the input functions *u* in *w* = col(*u*, *y*), which is defined as follows. A vector *u* = *u*(1), *u*(2), ..., *u*(*T*) is said to be persistently exciting (PE) of order *L* if matrix H*L*(*u*) has full row rank.

Assume that *u* is PE of at least order *L*, where *L* equals the sum of the number of inputs plus the state-space dimension (please check Theorem 1 in [26]); out of this, colspan(H*L*(*w*)) represents the set of all possible solutions of (2). That is, if the input is PE, then the complete dynamics of the electrical system can be fully described by the set of available measurements. While a model is able to determine all the possible outcomes of an electrical system as the solution of linear difference or differential equations, we are able to do the same by considering the linear combination of the row vectors of the Hankel matrix H*L*. Consequently, this array of data in a matrix owns the same model information.
