## 6 Illustrative Examples
### Example 1--Affine Linear Uncertainty Structure  
First, suppose the family of FO polynomials defined by the combination of the uncertain polynomial with affine linear uncertainty structure (inspired by its IO version in [10]):  
\[p_{LAF}(s,q)=s^{3,1}+(2q_{1}+q_{2}+2)s^{2,2}+(q_{1}+2q_{2}+1)s^{0,9}+(q_{1}+q_ {2}) \tag{19}\]  
and two kinds of uncertainty bounding set:  
1. Weighted \(L_{2}\) norm (an ellipsoid, or actually an ellipse in this two-dimensional case): \[ \begin{array}{l}\|q-q^{0}\|_{2,W}\leq 1\\ q^{0}=[2,2]\\ W=\left[\begin{array}{cc}w_{1}^{2}&0\\ 0&w_{2}^{2}\end{array}\right]=\left[\begin{array}{cc}1&0\\ 0&2\end{array}\right]\end{array} \] (20) that is, equivalently: \[ \begin{array}{l}\left(q_{1}-q_{1}^{0}\right)^{2}+2\left(q_{2}-q_{2}^{0} \right)^{2}\leq 1\\ q^{0}=[2,2]\end{array} \] (21)
2. \(L_{\infty}\) norm (a box, or actually a rectangle in this two-dimensional case): \[ \begin{array}{l}q_{1}\in[1,3]\\ q_{2}\in[1.5,2.5]\end{array} \] (22)  
The two-dimensional elliptical uncertainty bounding set (20), (21), with two uncertain parameters \(q_{1}\) and \(q_{2}\) that may vary within given bounds, is visualized in Figure 2, which shows not only the boundary (blue curve) but also 1000 randomly chosen internal points. Moreover, the rectangle formed by the displayed axes in Figure 2 concurs with the rectangular uncertainty bounding set (22). The value set of the ellipsoidal family of FO polynomials with affine linear uncertainty structure given by (19) and (20) for the frequency \(\omega=1\) is shown in Figure 3. The horizontal axis, labeled as \"Real Axis\", represents the real parts of the value set generally defined by (13), and the vertical axis, labeled as \"Imaginary Axis\" represents the imaginary parts of this value set. The same notation is also used in all the following relevant figures. The plotted value set consists of its boundaries, which are related to the boundaries of the elliptical uncertainty bounding set (20) from Figure 2, and the images of 1000 randomly selected points within this elliptical uncertainty bounding set. As can be seen, all internal points from the parameter space remain inside the value set in the complex plane as well. In other words, the boundaries of the value set are mapped only from the boundaries of the uncertainty bounding set (in the parameter space) for the studied case of the ellipsoidal family of FO polynomials. However, it should be noted that some parts of the boundaries in the uncertain parameter space can generally (in more than two-dimensional case) be mapped into the interior of the value set--see the example at page 142 in [5] where an edge of an uncertain cuboid is mapped to the inside the value set, and see also the Section 6.3 (Example 3). Anyway, it still holds true that the value set boundaries are mapped only from the uncertainty bounding set boundaries. The abovementioned results are valid for the affine linear uncertainty structure (and simpler \"independent\" one), but they need not be true for more general and more complex uncertainty structures as will be shown later. Furthermore, Figure 4 compares the value set of the ellipsoidal version of the family (19), (20) (blue solid curve) with the value set of the classical box version of the family (19), (22) (black dashed curve)--both for \(\omega=1\). The same comparison but for the value sets in the range of frequencies from 0 to 2.2 with the step 0.1 is shown in Figure 5 (the value sets would reach the fourth quadrant for higher frequencies since the highest order in (19) is 3.1). In compliance with the zero exclusion condition, which is described in the previous Section 5, both ellipsoidal and box version of the family of FO polynomials are robustly stable because all preconditions including the existence of a stable member are fulfilled and the complex plane origin (zero point) is excluded from the plotted value sets. Figure 2: Elliptical uncertainty bounding set (20) with 1000 randomly chosen internal points.