Algorithms of symbolic computations based on root trees for evaluating the capability of the control process

The article presents the methods and algorithms for evaluating management capabilities. These methods also determine the possibility of bringing the controlled system to a given point y * in the phase space. A set of reachability is a set of points in the phase space into which it is possible to move along the trajectory of the controlled system. It is necessary to check whether the point y * is included in the reachable set. Such problems are often encountered in practice, for example, when assessing the maneuverability of aircraft. Estimates of maneuverability should be accurate enough and implemented in a short time. The algorithms described in the article are based on the symbolic forms of solutions. When these algorithms are implemented, numerical data and character data are processed, and also are stored. To evaluate the ranges of values of expressions constructed by formulas, the article explores the structures of character data, including formal linear combinations of root labeled trees. Operations on the root tagged subtrees are performed by nesting the data structure in an associative data set. Reductions of the members of symbolic formulas are possible, if non-commutative operators are written through permutation operators. Such algorithms reduce the time of computation of operators in which derivatives are present, often exponentially. The article gives the examples of the application of these methods. Among these examples, we can mention the “Dobbins machine” control model, which represents the system of ordinary differential equations of the third order and describes the motion of an unmanned aerial vehicle (UAV). Since the flight speeds are in the range of 30-65 km/h, which is typical for UAVs, and the wind speed at an altitude of 30-200 m above the Earth level almost always exceeds 18 km/h, UAVs must effectively maneuver in the air flow. This allows us to examine the boundaries of reachable sets at each instant of time. An analysis of the reachable sets provides useful information for evaluating management capabilities