In the notation of Theorem 1, checking stability of an equilibrium in a circuit to a small current perturbation at node $k$ means finding out whether $\mathfrak{r}=0$ or not for the partial frequency response $G\!\left(s\right)$ from node $k$. Determining whether a realistic circuit is stable therefore boils down to determining whether it has poles in the complex right half-plane. Of course, such a clear-cut answer is hard to make from simulations of $G(j\omega)$ at finitely many points of the imaginary axis.

The partial frequency response of a realistic circuit with distributed elements is expected to have an infinite amount of poles as is the case for most delay systems [5]. The speed of approximation by rational functions to transfer functions of delay systems is rather low [13][14][15], hence high order models are typically needed to reach good accuracy on a broad frequency interval. However, when the degree goes large, rational approximation techniques based on interpolation which are often favored by electronics engineers are known to generate spurious poles whose physical interpretation is uneasy [16][17]; in fact, the extent to which the singularities of a rational approximant indicate those of the approximated function is a longstanding issue in approximation theory that cannot be answered independently of the approximation method one is using.

Theorem 1 suggests that identification methods should favor in this case a model class consisting of meromorphic functions with prescribed number $n$ of poles in the right half-plane, because the theoretical response is of this type with $n=N$. This seems better suited than trying to fit a rational approximant with free poles to the non-rational function $G\left(s\right)$. Two approximation techniques appear to be of special interest in this connection. The first is the half-plane version of the Adamjan-Arov-Krein theory on meromorphic approximation with $n$ poles in the uniform norm, also known as Hankel norm approximation, which is of standard use today in control and order reduction [18][19]. The second is best meromorphic approximation with $n$ unstable poles in $L^{2}$ of the line, which is equivalent to $H^{2}$-best rational approximation on the disk [20] for which efficient algorithms exist [21]. One would typically use this kind of approximation for increasing values of $n$: the case $n=0$ gives an estimate of the size of the unstable part, while the case $n=N$ (of course $N$ is unknown) would in principle allow one to recover $\mathfrak{r}$. Note that both algorithms work in the matrix-valued setting, which should be helpful to improve the estimation of $\mathfrak{r}$ by jointly approximating several partial frequency responses from the same node using a common denominator (cf. (2.6)) or even a block of partial frequency responses from a set of nodes to another set of nodes, using a matrix fractional representation for the block.

It is worth stressing that, at the functional level, computing $\mathfrak{r}$ is a linear operation. Assume indeed that the partial frequency response function $G\left(s\right)$ belongs to $L^{2}(j\R)$, where $j\R$ denotes the imaginary axis. This is a realistic assumption in that it is fulfilled as soon as the response rolls off like $1/|\omega|$ at infinity, which is typical of capacitive effects. Then, $G\left(s\right)$ will be stable if and only if it belongs to the Hardy space $\mathcal{H}^{2}$ of the right half plane. Similarly, let $\mathcal{H}_{-}^{2}$ denote the Hardy space of the left half-plane. Using the orthogonal decomposition

where $\oplus$ is the symbol for the direct sum of vector spaces , we see that the orthogonal projection of $Z(s)$ onto $\mathcal{H}_{-}^{2}$ is precisely $\mathfrak{r}$. In particular, the $L^{2}$-norm of the latter, as compared to the expected numerical error, provides one with an initial cheap test for instability [23][24]. When the response is not square summable but merely bounded on the imaginary axis, similar considerations are still valid in a non Hilbertian context.

Of course several basic issues remain to be addressed in practice. First of all, one has to extrapolate finitely many pointwise data for $G\!\left(j\omega\right)$ on a limited range of frequencies into a function given at all frequencies. In this connection, the behavior at infinity is an important question that requires special care, for instance working with weights. This is essential to estimate the unstable part, which is not a trivial task. Also, one may have to study the quantitative behavior of the response in greater detail to decide how significant this unstable part is with respect to numerical errors. Such questions are left here for further research, but the fact that $G\!\left(j\omega\right)$ may in principle be computed with good precision at a great many frequencies, unlike in most identification problems, leads the authors to believe that the problem is indeed amenable to function-theoretic techniques.