## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
### Test Results--Establishing the Optimal Dose \(x_{opt}\) of the Fertilizer  
The results of the research in the form of robustness distribution \(r(x)\) for uncertain vegetation states \((a_{2},a_{1},a_{0})\) for particular doses \(x\) of nitrogen fertilizer are presented in Figure 6.  
Robustness distribution \(r(x)\) provides some interesting information. Only above the dose \(x=48.091\) [kg/ha] there are such system states \((a_{2},a_{1},a_{0})\) where the yield \(y\) [q/ha] can (but does not have to) enter the desired tolerance corridor \([600,774]\) [q/ha]. For smaller doses of fertilizer \(x<48.091\), only yields below 600 [q/ha] can be obtained. Above the dose \(x=48.091\), the number of states \((a_{2},a_{1},a_{0})\) allowing to hit the tolerance corridor increases and reaches its maximum at the dose \(x=150\) [kg/ha]. At this dose, the robustness \(r(x)\) to system uncertainty is 0.488. This means that with over 48% of possible states \((a_{2},a_{1},a_{0})\) of the system, the yield will be in the range of \([600,774]\) [q/ha]. On the other hand, if one of the remaining 52% of possible states occurs in a given year, the desired yield will be not obtained. Although the result \(x_{opt}=150\) [kg/ha] does not guarantee the desired yield \(y\in[600,774]\), it is useful and realistic. There is no such dose \(x\) of fertilizer, which would guarantee a high yield in the range of \([600,774]\) [q/ha] under all possible growing conditions. If the growing conditions in the given year are critically unfavorable, e.g., very high temperatures, no rain  
Figure 6: Distribution of robustness \(r(x)\) of fertilizer doses \(x\) [kg/ha] to uncertain vegetation states \((a_{2},a_{1},a_{0})\) of sugar beet cultivation determined with the numerical method. at all, etc. it will not be possible to obtain high yields. Optimal fertilization promotes high yields, but does not guarantee them. This means that the problem under consideration does not have a tolerant solution in the \(\{0,1\}\) sense of [PERSON]'s definition [44]. There are no doses of fertilizer characterized by full (ideal) robustness \(r(x)=1\). There are only fertilizer doses \(x\) that have a fractional (realistic) robustness. The presented results (maximum fertilization robustness \(r(x)=0.488\) for \(x_{opt}=150\)) were obtained by the numerical method and they can be not perfectly accurate. It seems that more accurate results can be obtained using the geometric calculation method. Figure 7 shows the full 3D-space \(A_{2}\times A_{1}\times A_{0}\) of possible states \((a_{2},a_{1},a_{0})\) of the vegetation system corresponding to the control value (decision variable) \(x=150\) [kg/ha]. At some points of the possible state box in Figure 7, the yield values \(y\) corresponding to these points are given. With the known control value \(x=150\), the value of the yield \(y\) is linearly dependent on values of the coefficients \(a_{2},a_{1},a_{0}\) according to the equation of the system: \(y=a_{2}150^{2}+a_{1}150+a_{0}\). All points corresponding to the yield \(y=600\) [q/ha] can be determined in the box. These points correspond to the plane crossing the state box shown in Figure 7. The shaded part of the box is the space with the volume \(V_{r}\), that is the space of states \((a_{2},a_{1},a_{0})\) guaranteeing that the crop will hit the tolerance corridor [600, 774]. In the unshaded part of the box there are states \((a_{2},a_{1},a_{0})\) that do not hit the tolerance corridor. After calculating the volume \(V_{r}\) and \(V_{c}\) (volume of the whole box), it is easy to calculate the robustness \(r(x=150)=V_{r}/V_{c}=0.4875\). This robustness is not much different from the robustness \(r(x=150))=0.488\) determined by the numerical method. Both methods indicate \(x=150\) [kg/ha] as the optimal fertilization dose. However, attention should be paid to the fact (Figure 6) that fertilization doses lower than \(x=150\) also give high robustness to the uncertainty of the vegetation system.