## 3 Sub-Optimality of Algorithms 1-4 and Beyond Geometric Distribution
It is well known that model (43) is stable if and only if all the eigenvalues of matrix \(A\) belong to the stability domain \(\mathsf{D}_{\mathrm{s}}\) defined by:  
\[\mathcal{D}_{\mathrm{s}}=\Big{\{}s=\rho e^{i\theta}\in\mathsf{C}\,/\,|\theta| >\frac{\pi}{2}\Big{\}}. \tag{44}\]  
To explain how the stability domain is transformed by approximation (15) in Algorithm 3 or (18) in Algorithm 4, it is assumed that all the eigenvalues of model (43) are distinct. Thus, there exists a change of variable \(z(t)=P\,x(t)\), such that model (43) can be written as:  
\[\left\{ \begin{array}{c}\frac{d^{x}}{dt^{x}}z(t)=A_{P}z(t)+B_{P}u(t)\\ y(t)=C_{P}z(t)+D_{P}u(t)\end{array} \right., \tag{45}\]  
where matrix \(A_{P}\) is diagonal and can be written \(A_{P}=diag(\lambda_{i})\), \(i\in[1\ldots n]\). Using the Laplace transform, the characteristic equations of model (45) are thus  
\[s^{n}-\lambda_{i}=0,\,i\in[1\ldots n]. \tag{46}\]  
with approximation (15) or (18), these characteristic equations become:  
\[C_{0}\frac{\prod\limits_{k=1}^{N}\Big{(}1+\frac{s}{\omega_{k}^{\prime}}\Big{)} }{\prod\limits_{k=1}^{N}\Big{(}1+\frac{s}{\omega_{k}}\Big{)}}-\lambda_{i}=0 \qquad\text{and}\qquad C_{0}s\frac{\prod\limits_{k=1}^{N}\Big{(}1+\frac{s}{ \omega_{k}^{\prime}}\Big{)}}{\prod\limits_{k=1}^{N}\Big{(}1+\frac{s}{\omega_{ k}}\Big{)}}-\lambda_{i}=0. \tag{47}\]  
After approximation, model (43) is stable if characteristic Equation (47) has no root in the right half part of the complex plane. To evaluate the location of these roots, the Cauchy argument principle can be used. For that, the path \(\Gamma\) in Figure 9 can be used. Figure 9: Considered path \(\Gamma\). Images of path \(\Gamma\) by approximations (15) and (18) are represented by Figure 10. They are compared with the image of this path by s\({}^{\
u}\). From these curves, the stability domain of model (43) can be deduced. The image of path \(\Gamma\) by relation (47) is the image of path \(\Gamma\) by relations (15) and (18) translated by the vector \(\overrightarrow{V}_{\lambda_{i}}\) that appears in Figure 10 and that links the point of the coordinate (Re(\(\lambda_{i}\)), Im(\(\lambda_{i}\))) to the origin of the complex plane. Thus, after this translation the origin of the complex plane is located at the point (Re(\(\lambda_{i}\)), Im(\(\lambda_{i}\))). As the denominator of relations (47) has no root inside the path \(\Gamma\), if the origin (after translation) of the complex plane is inside one of the images, then according to argument principle, the corresponding characteristic equation has one root inside the path \(\Gamma\) and the system (43) is unstable. This permits the following theorem. **Theorem 1**. : _The stability domain in the complex plane, for the eigenvalues of matrix \(A_{P}\) (or for the poles) of model (45) or equivalently for model (43) is_  
* _outside the domain defined by,_ \(C_{0}\frac{\prod\limits_{k=1}^{N}\left(1+\frac{s}{\omega_{k}^{\prime}}\right)} {\prod\limits_{k=1}^{N}\left(1+\frac{s}{\omega_{k}^{\prime}}\right)}\)_,_ \(s=j\omega\)_,_ \(\omega\in[-\infty,\infty]\) _for approximation (_15_)_
* _outside the domain on the right of the curve_ \(C_{0}\frac{\prod\limits_{k=1}^{N}\left(1+\frac{s}{\omega_{k}^{\prime}}\right)} {\prod\limits_{k=1}^{N}\left(1+\frac{s}{\omega_{k}^{\prime}}\right)}\)_,_ \(s=j\omega\)_,_ \(\omega\in[-\infty,\infty]\)_, for approximation (_18_)._  
These stability domains are illustrated by Figure 11.  
This analysis also highlights stability issues that can occur after approximation. Considering Figure 12, which is an enlargement of Figure 11, it can be seen that some instability domains for model (43) after approximation are inside the stability domains for model (43) (before approximation). Some of these areas are marked with crosses on this figure. As a conclusion, approximation of a stable model using approximations (15) or (18) can produce an unstable model if some eigenvalues are located outside the intersections of the various stability domains.