## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
The values of \(x=-1.58\) and \(x=1.58\) are therefore robust at the degree 0.6 to the whole uncertainty range of \(a\), see \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]0 Tolerant control can therefore be implemented in this case, although there is no absolute certainty to realize it. Such a situation exists in many real systems. Therefore, it is fully realistic and the control in conditions of incomplete robustness to uncertainty can be called realistic control. The reason for the frequent failure to obtain an ideal tolerant control is the large number of uncertain parameters present in the systems, often a large range of their uncertainty, and high accuracy requirements for the output value \(y\) (small  
\[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]1 Illustration of the relationship between the possible roots \(x_{1}\) and \(x_{2}\) and the values of the uncertain coefficient \(a\) in the system control task \([2,4]x^{2}=[7,10]\). range of permissible error). In the system \([2,4]x^{2}=[4,20]\) the tolerance range \([4,20]\) is much larger than in the system \([2,4]x^{2}=[7,10]\). Both considered systems are a simplified version of the system \([\underline{a}_{2},\overline{a}_{2}]x^{2}+[\underline{a}_{1},\overline{a}_{1} ]x+[\underline{a}_{0},\overline{a}_{0}]=[\underline{y}_{T},\overline{y}_{T}]\). In the full version of the quadratic equation, there are not one, but three uncertainty generators: \([a_{2}]\), \([a_{1}]\), \([a_{0}]\). The greater the number of these generators, the smaller the chance of implementing the ideal tolerance control ensuring the certainty of hitting the tolerance corridor \([\underline{y}_{T},\overline{y}_{T}]\). Next, the tolerance control algorithm of the system described by the full quadratic equation will be presented. Algorithm for Determining the Tolerant Control of the System \([a_{2}]x^{2}+[a_{1}]x+[a_{0}]=[y_{T}]\)  
There is given a static system with an input \(x\) and an output \(y\) realizing the crisp relationship (20). \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]2  
The knowledge about coefficients of the system is uncertain:  
\[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]3  
The purpose of the search is to determine one control value \(x\) or a set of those values that will allow to obtain the output value \(y\) included in the tolerance corridor (22). \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]4  
The real values of coefficients \(a_{2}\), \(a_{1}\), \(a_{0}\) that will appear in a real system are independent from us: they depend on various external factors. They may be called system uncertainty generators (UGs). The intervals \([a_{2}]\), \([a_{1}]\), \([a_{0}]\) are epistemic, i.e., only one value is true. Whereas the tolerance corridor \([\underline{y}_{T},\overline{y}_{T}]\) is usually defined by the system user depending on his requirements, e.g., technological ones. The task is to determine such a control value \(x_{opt}\) (or a set of these values) that will allow to obtain the maximum control robustness for the uncertainty of the coefficients \(a_{2}\), \(a_{1}\), \(a_{0}\). Following steps should be realized to solve the problem. _Step 1_. Transferring the problem to the RDM variables space. _Step 2_. Checking whether the tolerance corridor \([\underline{y}_{T},\overline{y}_{T}]\) has been realistically chosen. If it is impossible to obtain results contained in the chosen tolerance corridor, then it should be corrected. _Step 3_. Determining the span \(SP_{x}\) of the search range \(X=[\underline{x},\overline{x}]\) of the control (decision) variable. \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]5 The distribution of the robustness \(r(x)\) of possible values of the tolerant control \(x\) in the system \([2,4]x^{2}=[7,10]\). Only partially robust control is possible. _Step 4_. Discretization of the span of variable \(x\). _Step 5_. Determining the distribution of the robustness \(r(x)\) for possible values of control variable \(x\) and determining its optimal value \(x_{opt}\) with the highest robustness.