## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance
Determining the Tolerant Solution of the Quadratic Interval Equation on the Example of Fertilizing Sugar Beet Cultivation  
Sugar beetroot is an important crop throughout growing regions of many countries. The root of the beet has a sugar content of approximately 17%. In Poland the profit from sugar beetroot cultivation starts at yield \([350,400]\) [q/ha]. Profitable sugar beet production depends on adequate soil fertility. Using appropriate agotechnical means some growers receive yield even \([600,800]\) [q/ha]. Nitrogen (N) is the most yield influencing nutrient and is critical to obtain maximum sugar beet yield and quality. The most common response curve for nitrogen fertilization is shown in \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]6  
Note that the field characteristic in Figure 5 is only an exemplary one. Each field has more or less different characteristics. In addition, the characteristics of one and the same field changes each year depending on the weather situation, insolation, rainfall, average temperatures in the area of cultivation, and others. A field with fertile soil (e.g., black earth) will require lower doses of fertilizer than a sandy field. Zone I in Figure 5 is a low fertilization zone. Zone III is a high fertilization zone. Zone IV is the zone of 'optimal' fertilization, which, however, is a bit dangerous. The reason: due to the lack of precise knowledge of the field characteristic, we can give too much fertilizer, which will reduce the possible yield. We will also waste some expensive fertilizer. Zone V is a zone of excessive fertilization. It leads to a reduction in harvest, excessive saturation of the soil with fertilizer and wastage of fertilizer. It can be said that plants 'do not like' too much fertilizer in the soil, just as humans do not like too much sugar in tea. The dependence of the yield \(y\) [q/ha] on the fertilization \(x\) [kg/ha] is approximated by the quadratic model (23). \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]7  
At the first glance, the model (23) seems theoretical and abstract. However, all its coefficients have an understandable and practical sense. The coefficient \(a_{0}\) [q/ha] means the yield obtained without fertilization \((x=0)\), i.e., obtained by the force of the soil itself. The coefficient \(a_{1}\) [q/kg], with a good approximation, means the yield increase caused by a fertilizer dose of only \(x=1\) [kg/ha]. It can also be called the influence of the first kg of fertilizer. Each next kg of fertilizer causes less and less yield gain. Reducing the impact of successive kilograms of fertilizer is determined by the coefficient \(a_{2}\) [q ha/kg\({}^{2}\)], which can be called the fertilizer inhibition factor. It has a negative value. To identify the values of the coefficients \((a_{2},a_{1},a_{0})\), we need to measure the variables \(x\) and \(y\) over a period of much more than 3 years. The values of these coefficients are uncertain as the actual impact of nitrogen fertilization changes each year. It depends on the quality of the fertilizer used, the degree of insolation of the field in a given year, the temporal distribution of this insolation, the sum of rainfall, the temporal distribution of rainfall, the  
\[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]8 The typical response curve \(y=f(x)\) for nitrogen fertilization \(x\). quality of the seeds used, and others. An exemplary interval model of the \(y\)-yield of a given field is given by the Formula (24). The value \(x\) is the fertilizer dose [kg/ha]. \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\]9 \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\]0  
Now suppose that a farmer, a field owner, knowing about all the uncertainties in beet vegetation, will be satisfied with the yield if it is within \(y\in[600,800]\) [q/ha]. These limits are the tolerance corridor of the expected, satisfying result. They depend, of course, on the farmer. The tolerance corridor can be compared to a target. The larger the target, the easier it is to hit.