# Methodology for monitoring the technical condition of onboard launch vehicle systems based on the processing of rapidly changing

DOI: 10.34759/trd-2021-121-19

### Аuthors

Uryupin I. P.

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

e-mail: uryupin93@yandex.ru

### Abstract

The purpose of the research consists in developing an optimization algorithm for continuous systems in the class of piecewise constant controls. In classical problems of optimal control of continuous systems, the admissible controls are, as a rule, bounded measurable ones (in applied problems, they are piecewise continuous). In some complex continuous systems, physical implementation of such admissible control seems impossible Then one can narrow down the set of admissible controls, for example, to the class of piecewise constant controls with a fixed number of switchings and search for a solution in this narrow class. It is clear, the suboptimal controls herewith will be obtained, which, however, with an unlimited increase in the number of switchings, will tend to the optimal one. Thus, the problem of minimizing the number of switchings becomes actual.

To solve the set problem, a numerical-analytical algorithm has been developed based on the necessary conditions for optimality of switched systems. Boundary problems and formulas expressing the optimal values of piecewise constant control are obtained analytically, according to the necessary conditions. The resulting system of equations, as a rule, is transcendental and its solution is rather difficult. Thus, the author proposes to use the numerical minimization of the functional, followed by fulfillment verification of the necessary conditions to search for the optimal switching moments. The proposed algorithm is realized in MATLAB.

The problem of a linear oscillator controlling with quadratic functional is being considered as an example. The optimal solution to this problem in the class of continuous controls was obtained using the maximum principle. In this work, the optimal solution to the problem in a narrower class of piecewise constant controls is being searched for. This solution is being found using necessary conditions for the switched systems optimality. It may be regarded as a solution close to optimal continuous control. Besides the problem of an optimal piecewise constant control synthesizing, the problem of finding the minimum number of switchings, at which the difference between the approximate solution and the exact one does not exceed a given error is also solved.

The main result of the work is a numerical-analytical algorithm for minimizing the number of switchings of optimal piecewise-constant controls for an approximate solution of the optimal control problem for continuous systems.

### Keywords:

theory of control, switched systems, switching minimization, optimization, algorithm, hybrid systems

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