Synthesis of piecewise-smooth approximations of aircraft movement trajectories

System analysis, control and data processing


Uryupin I. P.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The goal of the study consists in developing an algorithm for optimal switched systems synthesis. Switched systems relate to hybrid control systems and described by differential equations and recursion inclusions. They serve as mathematical models of dynamic systems with an automata part. Switched systems find wide application in aerospace engineering in the field of flying vehicles movement control. The article envisages the problem of optimal trajectory synthesis of the switched system, approximating a continuous trajectory of the aircraft movement. The trajectory is a

The trajectory is a piecewise-linear function and, in general, it cannot coincide with the predetermined continuous curve. Solution of the set problem allows develop and check a method of synthesis of optimal piecewise smooth approximations of movement trajectories which can be employed in complex applied problems.

The algorithm is based on sufficient conditions of optimality. The synthesis technique consists in finding generating functions of price, conditional optimal structures and conditional optimal switching moments from which the price function is compiled (the Hamiltonian – Jacobi – Bellman function). The synthesized optimal positional structure of the switched system allows receiving optimal trajectories for any initial states.

The result of this work is the solution of the problem of synthesizing optimal piecewise smooth approximations of the aircraft movement trajectories, as well as the method applied herewith of creating generating functions of the price and an optimal positional structure. The developed method may be impemented in Aerospace area.

The problem solved in the article can be employed as a test task for theoretical research in the field of optimal control. The developed algorithm of creating generating function of the price and a positional structure for the trajectories piecewise smooth approximations can be applied to solve other more complex problems. Given, that even in academic examples the solution is numerical, the application of the proposed synthesis methods is inextricably linked with the development of appropriate approximate algorithms, as well as programs for the numerical implementation of these algorithms.


switched system, synthesis, optimal trajectory, algorithm, optimization, price function, piecewise–linear approximation


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