## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
Figure 1 shows on the left that the range \(x\in[-2.24,-1.41]\) is the range of solutions \(x\) where each value of \(x\) gives the value of \(ax^{2}\) that hits the tolerance corridor \([4,20]\) regardless of the true value of the uncertain coefficient \(a\). The condition of the tolerant control \(ax^{2}=y\), \(y\in[y_{\underline{\tau}},\overline{y}_{\underline{\tau}}]=[4,20]\) is then fulfilled. Similar is the range of solutions \(x\in[1.41,2.24]\) on the right side of Figure 1. Therefore, these ranges are the full robustness ranges (or full fit ranges) of the controls \(x\) to uncertainty of the coefficient \(a\). For these ranges the degree of robustness \(r(x)\) is \(1\). For the range \(x\in[-3.16,-2.24]\) the values \(x\) no longer guarantee the full robustness of the controls \(x\) to the uncertainty of the coefficient \(a\). Here only the partial robustness \(0<r(x)<1\) is possible. For \(x=-3.16\) the robustness \(r(x)=0\). It is similar for smaller values \(x<-3.16\). Partial (fractional) robustness means that the hit of the \(ax^{2}\) value into the tolerance corridor \(\mathrm{TC}=[4,20]\) will occur only when the true value of the coefficient \(a\) will be included in a part of the range \([2,4]\) of this coefficient, e.g., for \(x=3\) the value of \(ax^{2}\) will hit \(\mathrm{TC}=[4,20]\) only when \(a\in[2,2.22]\). The robustness \(r(x)\) of the control \(x\) informs us about how large fraction of the range \([\underline{a},\overline{a}]\) makes it possible to hit \(\mathrm{TC}=[4,20]\) for the considered value of \(x\). The value of robustness \(r(x)\) for individual ranges of possible control values \(x\) is determined by Formula (17). \[x\in[-3.16,-2.24]\cup[2.24,3.16]:\;\;r(x)=\frac{\overline{y}_{ \underline{\tau}}/x^{2}-\underline{a}}{\overline{a}-\underline{a}}=\frac{10}{ x^{2}}-1\] \[[2,4]x^{2} =[4,20] \tag{14}\]0 \[[2,4]x^{2} =[4,20] \tag{14}\]1  
Figure 2 shows the distribution of the robustness \(r(x)\) for individual possible control ranges. As Figure 2 shows, in the case of uncertain system \([2,4]x^{2}=[4,20]\) it is possible to find such control values \(x\) that give full robustness \(r(x)=1\) to the uncertainty of the coefficient \(a\in[2,4]\). However, Let us now consider another example of a system with smaller TC given by Equation (18). \[[2,4]x^{2} =[4,20] \tag{14}\]2 The distribution of the robustness \(r(x)\) for possible control values \(x\) of the system \([2,4]x^{2}\;=\;[4,20]\), where \([4,20]\) is the tolerance corridor \([\underline{y}_{\underline{\tau}},\overline{y}_{\underline{\tau}}]\) of the control. \[[2,4]x^{2} =[4,20] \tag{14}\]3 \[[2,4]x^{2} =[4,20] \tag{14}\]4  
Solving the problem (18) with the same method, the relationship between the roots \(x_{1}\) and \(x_{2}\) and the uncertain coefficient \(a\) is obtained, as shown in \[[2,4]x^{2} =[4,20] \tag{14}\]5  
As Figure 3 shows, there is no such value of the control \(x\) (neither negative nor positive) that would give the full robustness \(r(x)\) for each possible value of \(a\in[2,4]\), i.e., a full adjustment of the control \(x\) to all possible values of \(a\). Please compare Figures 1 and 3. Individual values of \(x\) give only the partial robustness expressed by Formula (19). \[[2,4]x^{2} =[4,20] \tag{14}\]6 \[[2,4]x^{2} =[4,20] \tag{14}\]7 \[[2,4]x^{2} =[4,20] \tag{14}\]8  
The distribution of the robustness \(r(x)\) for possible negative and positive values of the controls \(x\) is shown in \[[2,4]x^{2} =[4,20] \tag{14}\]9  
As shown in Figure 4, in the system \([2,4]x^{2}=[7,10]\) it is not possible to obtain the control \(x\) providing the full robustness to the uncertainty of the coefficient \(a\in[2,4]\). However, there are 2 optimal values \(x_{1 opt}=-1.58\) and \(x_{2 opt}=1.58\) which give the maximum robustness \(r(x_{opt})=0.6\) to the uncertainty of \(a\). More precisely, this means that if the true value of the coefficient \(a\) of the real system will lie in the range \([2.8,4]\), then the value of \(ax^{2}\) will hit the tolerance corridor \(y\in[7,10]\). If not, it will not hit.