## 4 Uncertainty Bounding Set  
An a priori bound \(Q\) for the vector of real uncertain parameters \(q\) is assumed for systems with parametric uncertainty. This \(Q\) is called the uncertainty bounding set and it is supposed as a ball in an appropriate norm. The typical shapes of \(Q\) are a box (for \(L_{\infty}\) norm), a sphere or an ellipsoid (for \(L_{2}\) norm or weighted \(L_{2}\) norm, respectively), and a diamond (for \(L_{1}\) norm). [5; 10; 11; 15]. The most common case uses the \(L_{\infty}\) norm [5; 10; 11; 15]:  
\[\left\|q\right\|_{\infty}=\max_{i}\left|q_{i}\right| \tag{7}\]  
and a ball in this norm is referred to as a box. In practice, the box is given by its components, that is, by the real intervals determining the minimal and maximal bounds for the individual uncertain parameters. The second important case utilizes the \(L_{2}\) norm or weighted \(L_{2}\) norm [5; 10; 11; 15]. Standard Euclidean norm is:  
\[\left\|q\right\|_{2}=\sqrt{\sum_{i=1}^{n}q_{i}^{2}} \tag{8}\]  
and the weighted Euclidean norm has the form:  
\[\left\|q\right\|_{2,W}=\sqrt{q^{T}Wq} \tag{9}\]  
where \(q\in\mathbb{R}^{k}\) and \(W=\text{diag}\Big{(}w_{1}^{2},w_{2}^{2},\ldots,w_{k}^{2}\Big{)}\) is a positive definite symmetric matrix of size \(k\times k\) that constitutes the weighting matrix. Assuming \(r\leq 0\) and \(q^{0}\in\mathbb{R}^{k}\), the ellipsoid in \(\mathbb{R}^{k}\) centered at \(q^{0}\) can be defined by the inequality:  
\[\Big{(}q-q^{0}\Big{)}^{T}W\Big{(}q-q^{0}\Big{)}\leq r^{2} \tag{10}\]or equivalently:  
\[\left\|q-q^{0}\right\|_{2,W}\leq r \tag{11}\]  
For example, the special case of the two-dimensional ellipse of uncertain parameters (\(k=2\)) can be readily visualized for:  
\[W=\left[ \begin{array}{cc}w_{1}^{2}&0\\ 0&w_{2}^{2}\end{array} \right] \tag{12}\]  
The corresponding ellipse is shown in Figure 1[5, 15]. It is worth to discuss the choice of the \(Q\) shape, i.e., if a box or an ellipsoid should be preferred for a given problem. In a number of engineering applications, the variations of the real physical parameters mutually independent, and consequently, it is natural to suppose \(Q\) as a box. Nevertheless, according to [5], the ellipsoids can be convenient under \"imprecise description\" of the uncertainty bounds, in other words, if the true \(Q\) lies between some minimally perturbed \(Q_{min}\) and maximally perturbed \(Q_{max}\) (both given by the boxes) and an ellipsoid can interpolate them. Moreover, the selection should respect available methods for the solution of the problem at hand [5]. Besides, as stated in [52], the mathematical models based on the physical principles typically have \(Q\) in the shape of a box, but the identification techniques mostly lead to \(Q\) in the shape of an ellipsoid (see also [20]). Robust Stability of Families of Fractional-Order Polynomials with Ellipsoidal Parametric Uncertainty  
It was already shown that the family of FO polynomials is described generally by (6). Obviously, this family is robustly stable if and only if \(p(s,q)\) is stable for all \(q\in Q\). In this paper, uncertainty bounding set \(Q\) is supposed to be an ellipsoid and the FO uncertain polynomial (5) is assumed to have some complicated parametric uncertainty structure (affine linear, multilinear, polynomial, or general). Thus, the resulting families of FO polynomials studied in this paper are called the ellipsoidal families of FO polynomials with complex uncertainty structure. From the robust stability viewpoint, there are two main complications that significantly reduce the number of suitable tools for analyzing such families: the complexity of the uncertainty structure, and FO. As mentioned above in the Introduction, many existing methods, such as the Soh-Barger-Dabke theorem [5, 28], Biernacki-Hwang-Bhattacharyya theorem [5, 29], Barmish-Tempo theorem [5, 30], and Tsypkin-Polyak criterion [31] are applicable to the IO families with relatively simple uncertainty structures.