## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
The maximum dose can be set on the basis of the previous Step as 180, or slightly more, e.g., 200 [kg/ha]. In the presented study, the span \(X=[0,200]\) [kg/ha] was assumed. _Step 4._ Discretization of the span of variable \(x\). The search step \(\Delta x=0.1\) was assumed, which means that the number \(n=2001\) of discrete values from the set \(X\) were analyzed:  
\[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]6  
_Step 5._ Determining the robustness \(r(x_{i}^{d})\) of successive discrete values of the control variable \(x_{i}^{d}\in X^{d}\). Robustness calculations can be implemented either analytically using the geometric calculation method or numerically. We will come back to the analytical method in Section 4.1. However, since for three uncertain coefficients the exact calculations are very complex, it will be shown how to determine the approximate value of the robustness numerically. The coefficients of the equation can be discretized in a different or identical way. In our research, we used the same discretization to obtain the same number of discrete values for each coefficient. In the initial search, we used discretization for 11 values, in a more precise search for 51 values. Increasing the number of discrete values increases the chances of more accurately detecting the optimal dose of fertilization. The continuous set \(A_{2}=[-0.012,-0.010]\) was discretized into \(m_{a2}\) discrete values:  
\[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]7  
The continuous sets \(A_{1}\) and \(A_{0}\) were discretized in a similar way:  
\[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]8  
\[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]9  
Each triple \((a_{2,j},a_{1,k},a_{0,l})\) describes one possible state of an uncertain system. The number of all possible states that may occur is \(m_{a2}\cdot m_{a1}\cdot m_{a0}\). For each analyzed control value \(x_{i}^{d}\)\(m_{a2}\cdot m_{a1}\cdot m_{a0}\) output values \(y_{i}(x_{i}^{d},a_{2,j},a_{1,k},a_{0,l})\) corresponding to all possible combinations \((a_{2,j},a_{1,k},a_{0,l})\) of system states were calculated. Then, the sum \(p(x_{i}^{d})\) of these \(y_{i}\) values which hit the tolerance corridor \([\underline{y}_{T},\overline{y}_{T}]\) was determined, the condition (33). \[x_{1}(\gamma_{a},\gamma_{y})=-\sqrt{\frac{4+16\gamma_{y}}{2+2 \gamma_{a}}},\;\;\;x_{2}(\gamma_{a},\gamma_{y})=+\sqrt{\frac{4+16\gamma_{y}}{ 2+2\gamma_{a}}} \tag{15}\]0  
Next, for each analyzed value of \(x_{i}^{d}\) its robustness \(r(x_{i}^{d})\) can be calculated:  
\[x_{1}(\gamma_{a},\gamma_{y})=-\sqrt{\frac{4+16\gamma_{y}}{2+2 \gamma_{a}}},\;\;\;x_{2}(\gamma_{a},\gamma_{y})=+\sqrt{\frac{4+16\gamma_{y}}{ 2+2\gamma_{a}}} \tag{15}\]1Correctly calculated robustness satisfies the condition \(r(x_{i}^{d})\in[0,1]\). Short explanation: if e.g., each of the sets \(A_{2}\), \(A_{1}\), \(A_{0}\) is discretized into 51 values (\(m_{a2}=51\), \(m_{a2}=51\), \(m_{a0}=51\)), then the number of all possible discrete vegetative states of the system is \(51^{3}=132651\). Then, for each analyzed control value \(x_{i}\), it is necessary to calculate 132,651 values of the output \(y\) and determine the number of hits \(p(x_{i})\) in the tolerance corridor. At the end of the test, the optimal value \(x_{opt}\) [kg/ha] of the fertilizer dose corresponding to the maximum robustness \(r(x)\) of the decision value \(x\) should be determined. Since the calculations performed in robustness determining are very simple, the calculation time for \(x_{opt}\) did not exceed several minutes on an average quality computer, even with a very accurate discretization of the \(x\) variable and coefficients \(a_{2}\), \(a_{1}\), \(a_{0}\).