Numerical estimations of maximum deviations for aircraft trajectories in the atmosphere

In this article we studyguaranteedmethods for solving differential equations systems with control actions and its application for problems of aircrafts trajectories maximum deviations estimating. The problem statement of maximum deviations estimating is caused by the necessity to evaluate reliability of complexnonlinear controlled systems that operate under the influenceof perturbations. The exampleof such system is an aircraft: an airplane, a rocket, a spacecraft. We should emphasize the research of an airplane movement during an autoland approach among problems of maximum deviations estimating. This research gives an answer whether a violation of restrictions imposed on the kinematic parameters of an airplane touching down a runway is possible or not. The problems of limit deviations evaluating also includeestimating ofthe possibilityof an aircraft motion stability’s loss for a given time interval. The greatest difficultyin solving suchproblems arises inthe case when flight conditionsare not fixed, for example, when consideringthe descentinto the atmosphere of a spacecraft orbiter. In this case the loss of stability criteriathemselves are not formulated, if we do not confine ourselves to the simplest case of a linear system, forwhich theRouth-Hurwitz conditions can be used. Some threshold orcritical valueof one of theparameters of an airplane can serve as a simplisticcriterion of stability loss. For example, the angle of attack orslip angle can be taken as such parameter. Reachable sets(collections of all the trajectories of controlled systems) make a figure in such problems. These setsare used in problems of guaranteed or minimax estimationof solutions of dynamical systems ifexternal perturbations that influence on a system andobservation errorsare enclosedwithin a certain range(constrained by limitations). The analysis of works that estimatereachable sets indicatesthat the reliable estimationof reachable setsof controlled systems under uncertainty, if theright-hand sidesof these systemsdepend nonlinearly on thecontrol actions,is not always possible. Therefore the possibility of the guaranteed methods based on symbolic representation of solutions for reachable sets evaluation will be useful for specialists in control. The article presents theresults of the applicationof the numericalmethods basedon the constructionof symbolicformulas of solutions andevaluating all of its possiblevalues.