Realistic Optimal Tolerant Solution of the Quadratic Interval Equation and Determining the Optimal Control Decision on the Example of Plant Fertilization

Abstract

In scientific journals, it is increasingly common to find articles presenting methods for solving problems not based on idealistic mathematical models containing perfectly accurate coefficient values that cannot be obtained in practice, but on models in which coefficient values are affected by uncertainty and are expressed in the form of intervals, fuzzy numbers, etc. However, solving tasks with interval coefficients is not fully mastered, and a number of such problems cannot be solved by currently known methods. There is undeniably a research gap here. The article presents a method for solving problems governed by the quadratic interval equation and shows how to find the tolerant optimal control value of such a system. This makes it possible to solve problems that could not be solved before. The paper introduces a new concept of the degree of robustness of the control to the set of all possible multidimensional states of the system resulting from its uncertainties. The method presented in the article was applied to an example of determining the optimal value of nitrogen fertilization of a sugar beet plantation, the vegetation of which is under uncertainty. It would be unrealistic to assume precise knowledge of crop characteristics here. The proposed method allows to determine the value of fertilization, which gives a chance to obtain the desired yield for the maximum number of field conditions that can occur during the growing season.

Keywords

• interval arithmetic
• interval equations
• multidimensional interval arithmetic
• robustness to uncertainty
• plant fertilization modeling