adaptive Runge-Kutta method (as in [@chen2013refined]), or by any oversampling technique with intersample behaviour assumptions.

We now prove the consistency of the proposed estimator for the multisine input. The assumptions we use during the analysis are the following:

1.  \[assumption1\] The true system $B^*(p)/A^*(p)$ is proper ($n^* \geq m^*$) and asymptotically stable with $A^*(p)$ and $B^*(p)$ being coprime.

2.  \[assumption2\] The disturbance sequence $\{v(t_k)\}$ is a zero-mean stationary random process.

3.  \[assumption3\] The number of sinusoids of the input, $m_u$, satisfies $m_u\geq (n+m)/2$, and the input offset, $\alpha_0$, is different from zero.

4.  \[assumption4\] All the zeros of $A_j(p)$ have strictly negative real parts, $n\geq m$, with $A_j(p)$ and $B_j(p)$ being coprime.

5.  \[assumption5\] The degrees of the polynomials in the model satisfy $\min(n-n^*,m-m^*)=0$.

Assumptions \[assumption1\] and \[assumption2\] are standard. The condition in Assumption \[assumption3\] is a persistence of excitation requirement, where $\alpha_0 \neq 0$ is set only for simplicity in our derivations and can be removed.[^2] Given that the poles of unstable models are reflected in line 6 of Algorithm \[algorithm1\], Assumption \[assumption4\] is met in practice. Assumption \[assumption5\] takes into account the model structure, as it ensures a unique solution of the model parameters to be obtained.

Since 