## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
Thus, the task of calculations is to find the optimal dose of nitrogen fertilizer \(x\) [kg/ha] under the existing uncertainty of beet vegetation. This task is therefore a tolerant control defined by Formula (25). \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]1 \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]2  
All 3 vegetation coefficients (\(a_{2}\), \(a_{1}\), \(a_{0}\)) are uncertain. We do not know what their real value will be in the year under consideration. Hence, they can be called uncertainty generators (UGs) or uncertainty sources (USs) or simply model disturbances that make it difficult for us to solve the problem. In contrast, the tolerance corridor (TC): \([\underline{y}_{T},\overline{y}_{T}]=[600,800]\) is not UG. It does not make it difficult, but makes it easier for us to find a solution \(x_{opt}\) to the problem. Now, an important note. No UGs may be reduced from the system model during its transformations and performed arithmetic operations, e.g., addition, division etc. Each UG exists in a real system and creates uncertainty in its operation. For example, you cannot replace 2 or 3 UGs with one resultant UG. And this is how the SIA works, which recommends performing the addition operation as follows: \([1,3]+[2,5]=[3,8]\). This results in a loss of information. In general, this leads to results that are often partially wrong (inaccurate) or completely wrong. In a similar way, UGs are reduced from mathematical models by all 1-dimensional types of interval arithmetic whereby the result of operations on many intervals (UGs) is one interval (one UG). The only exception is the MIA, which retains all UGs in the form of RDM \(\gamma\)-variables in its multidimensional final result. The task of determining the optimal value of the tolerance control \(x_{opt}\) for QIE will be presented in the following steps. _Step 1._ Transfering of the problem to the RDM variable space. \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]3  
_Step 2._ Checking whether in the case of the equation under consideration the tolerance corridor is realistic, i.e., whether it is possible to obtain the results lying in the tolerance corridor at all. In the case of the fertilization problem, the goal is to obtain \(y\)-yields in the range of \([600,800]\) [q/ha]. This checking task is easy, because from Equation (25) it is easy to conclude that the maximum possible yield can only occur in the most favorable vegetation conditions corresponding to the maximum values of the coefficients \(a_{2}=-0.010\), \(a_{1}=3.6\), \(a_{0}=450\) (the fertilization \(x\) is always positive). These conditions are described by the Equation (27). \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]4  
Using the generally known Vieta formulas for solving the quadratic equation, we can find out that the maximum yield can occur with fertilization \(x=-\frac{a_{1}}{2a_{2}}=180\) [kg/ha]. The value of this yield can be calculated using the Formula (28). \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]5The result (28) means that it is not possible to obtain the yield value above 774 [q/ha]. For this reason, it is necessary to correct the tolerance corridor to the real range \([\underline{y}_{T},\overline{y}_{T}]=[600,774]\). This corridor will be used in the further calculations. _Step 3._ Determination of the span of the search range \(X=[\underline{x},\overline{x}]\) of the fertilizer dose (decision variable). After transforming the problem to the RDM variable space, we can easily solve the equation and determine the roots \(x_{1}(\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\) and \(x_{2}(\gamma_{a2},\gamma_{a1},\gamma_{a0},\gamma_{yT})\). We can next determine the span \(SP_{x1}\) and \(SP_{x2}\) in a similar way as in Formula (17). However, the span of \(X\) in practical problems can usually be determined on the basis of the knowledge of the problem expert. In the case of fertilizing plants, the minimum dose of fertilizer \(x\) is zero.