## 4 Uncertainty Bounding Set
However, there is a versatile graphical technique combining the value set concept with the zero exclusion condition [5] available. It fundamentally works not only for the complex uncertainty structures [7, 8] (including a class of retarded quasi-polynomials [53], anisochronic models with internal delays [32] and FO systems [9, 33, 34, 35, 36], but also for (IO) systems with ellipsoidal parametric  
Figure 1: Ellipsoidal geometry defined by weighted Euclidean norm [5, 15]. uncertainty [5; 10; 11; 15]. The robust stability of FO polynomials with a complicated structure of parametric uncertainty and box-shaped uncertainty bounding set by means of the universal graphical approach was studied in [9]. On the other hand, the ellipsoidal families of FO polynomials represent the topic of interest in this work. Given the family of polynomials (6), the value set at the frequency \(\omega\in\mathbb{R}\) is defined in [5] as:  
\[p(j\omega,Q)=\{p(j\omega,q):q\in Q\} \tag{13}\]  
i.e., \(p(j\omega,Q)\) is the image of \(Q\) under \(p(j\omega,q)\). In other words, the value set can be obtained by substituting \(s\) for \(j\omega\), fixing \(\omega\), and letting \(q\) range over the set \(Q\). The zero exclusion condition is formulated as [5]: Suppose that a family of polynomials (6) has an invariant degree with associated \(Q\) that is pathwise connected, continuous \(\rho_{i}(q)\) for \(i=0,1,\ldots,n\) and at least one stable member \(p(s,q^{0})\). Then the family (6) is robustly stable if and only if the origin (zero point) is excluded from the value set \(p(j\omega,Q)\) at all frequencies \(\omega\geq 0\), that is, (6) is robustly stable if and only if:  
\[0\
otin p(j\omega,Q)\quad\forall\omega\geq 0 \tag{14}\]  
It was derived in [5] that the value set for an ellipsoidal family of IO polynomials with independent parametric uncertainty structure can be expressed analytically as follows. Assume the ellipsoidal polynomial family with invariant degree \(n\geq 1\) described by:  
\[ \begin{split}& p(s,q)=p_{0}(s)+\sum\limits_{i=0}^{n}q_{i}s^{i}\\ & p_{0}(s)=p(s,q^{0})=\sum\limits_{i=0}^{n}a_{i}s^{i}\\ &\|q\|_{2,W}\leq r\\ & W=\text{diag}\Big{(}w_{1}^{2},w_{2}^{2},\ldots,w_{n}^{2}\Big{)} \end{split}  \tag{15}\]  
Then, the value set at each frequency \(\omega>0\) is given by an ellipse centered at the nominal \(p_{0}(j\omega)\) with major axis in the real direction having a length:  
\[R=2\gamma\left(\sum\limits_{even}w_{i}^{2}\omega^{2i}\right)^{1/2} \tag{16}\]  
and major axis in the imaginary direction having a length:  
\[I=2\gamma\left(\sum\limits_{i\omega d}w_{i}^{2}\omega^{2i}\right)^{1/2} \tag{17}\]  
Furthermore, for the degenerate event of \(\omega=0\), the value set is defined by the real interval:  
\[p(j0,Q)=\langle a_{0}-r,\,a_{0}+r\rangle \tag{18}\]  
Incidentally, these computations are implemented in the routine \"spherplot\" in the Polynomial Toolbox for Matlab (available from the version 2.5) [12]. Nevertheless, the application of this technique is limited to the IO case and independent uncertainty structure. In this paper, the computation and visualization of the value sets for the ellipsoidal families of FO polynomials with complex parametric uncertainty structure are based on random or fixed-step sampling (gridding) the uncertain parameters and on the subsequent direct calculation of the related partial points of the value sets for an assumed range of non-negative frequencies. These straightforward sampling approaches are used because it is generally very difficult to find the meaningful areas in the space of uncertain parameters a priori--e.g., the examples in the following Section 6 will demonstrate that some parts of boundaries of the value sets may be mapped from the internal points in the parameter space. The probabilistic methods in the analysis and design of systems subject to deterministic and stochastic uncertainty are discussed e.g., in [54; 55]. When the value sets are plotted, the robust stability can be readily checked with the necessary and sufficient condition.