Application of sensitive functions, which used to compute tubes including the trajectories of control systems

This article presented the use of sensitivity functions to compute the boundaries of inclusions of control systems reachability sets and their application to problems of estimation tolerances of aircraft motion, or missiles motion, or spacecraft motion. As a rule, the model of the control system is carried out throughout the range of the defining set of parameters in the framework of the sensitivity of numerical investigation of the parametric. The practical application of this approach is very often impractical or impossible because of the huge number of required computations and countless of the results. The combined use of the sensitivity functions and the analytical formulas of solutions proposed and implemented in the article, can effectively compute the inclusion of reachable sets. These sets include all trajectories of the control system, starting at the initial time point in the initial set. The inclusion of reachable sets are used in problems of guaranteed estimation of variance sets aircraft and in problems of control tolerances, considering that the current external disturbances of system and errors of observation are enclosed within certain limits (constrained by limitations). Defined sensitivity functions are derivatives of various state variables with respect the parameters of the appropriate group. Obtained these functions are solutions of the sensitivity equations constructed directly from a known parametric model of the system. Using the method, based on symbolical formulas for the solution and based on sensitivity function, allows getting a reliable estimate of reachable sets of control systems in conditions of uncertainty. Control actions are included on the right side of these systems arbitrarily, not only as an additive term. Application of this method involves the problem of estimating the maximum deviations of the aircraft motion at the stage of the automatic approach, the problem of determining the possibility of loss of stability of the aircraft motion at a given time, that is the problem of safety of the aircraft trajectory, the problem of the helicopter landing. Simplified criteria for buckling in such problems are the computation of a threshold or critical value of one of the motion parameters, and evaluation of the boundaries of all possible trajectories. The article presents the results of numerical methods based on the use of analytical formulas and sensitivity functions and evaluating all its possible values (reachable sets of control systems).