Bilinear interpolation of program control in approach problem

Abstract

We consider a controlled system involving a constant two-dimensional vector parameter, the approximate value of which is reported to the controlling person only at the moment of the start of movement. Apriori only the set of possible values of these unknown parameter is given. For this controlled system we pose the problem on approaching the target set at a given time. At the same time, we suppose that the controlling person has no the ability to carry out cumbersome calculations in real time associated with the construction of such resolving structures as reachability sets and integral funnels. Therefore, to solve this problem, it is proposed to calculate in advance several “node” resolving controls for parameter values, which are nodes of a grid covering a set of possible parameter values. If at the moment of the beginning of the movement, the parameter value turns out not coincide with any of the grid nodes, it is proposed to calculate the software control by using linear interpolation formulas. However, this procedure can be effective only if a linear combination of controls corresponding to the same “guide” is used in the terminology of the N.N. Krasovsky extreme aiming method. For the possibility of effective use of linear interpolation, it is proposed to build four “node” resolving controls for each grid node and, in addition, to use the method of dividing the control into the main and compensating ones. Due to the application of the latter method, the computed solvability set turns out to be somewhat less than the actual one, but the accuracy of translating the state of the system to the target set increases. A nonlinear generalization of the Zermelo navigation problem is considered as an example.