### Linear dynamic system phase coordinates vector:sensitivity functions definition according to the integro-power series from the matrix of parameter variations

**Published:**29.06.2022

**Authors:** Tushev O.N., Belyaev A.V.

**Published in issue: **#6(126)/2022

**DOI: **10.18698/2308-6033-2022-6-2185

**Category:** Aviation and Rocket-Space Engineering | **Chapter:** Design, construction and production of aircraft

The paper introduces a method for calculating sensitivity functions of the first and second orders of phase coordinates that do not require the integration of cumbersome chain-related systems of differential equations. The general and particular solutions of the vector differential equation of motion, written in the Cauchy form, are expressed in terms of the fundamental matrix. With the help of formal transformations, the vector of phase coordinates is represented as a convergent integro-power series with respect to the matrix that determines the variations of the elements of the matrix of system equation coefficients. In this work, the system is called the variation matrix. Then the relations obtained are transformed by special operations to an explicit form with respect to these variations up to and including the quadratic approximation. Within the framework of the matrix apparatus, there are similar expansions in terms of variations of the external influence and initial conditions. To exclude numerous "parasitic" operations of multiplication by zero in calculations, it is proposed to use special operations of matrix algebra. The original and inverse fundamental matrices are treated as multiplicative integrals, which ensures their simple calculation in time using recurrent formulas. The resulting apparatus is built entirely on matrix operations, which ensures simple machine implementation and versatility.

**References**

**[1]**Rozenvasser E.N., Yusupov R.M. Trudy SPIIRAN — SPIIRAS Proceedings, 2013, no. 2 (25), pp. 13–41.

**[2]**Bushuev A.Yu., Yakovlev D.O. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of the Bauman Moscow State Technical University. Series Natural Sciences, 2011, no. S3, pp. 66–69.

**[3]**Rozenvasser E.N., Yusupov R.M. Chuvstvitelnost sistem upravleniya [Sensitivity of control systems]. Moscow, Nauka Publ., 1981, 464 p.

**[4]**Hog E., Choi K., Komkov V. Analiz chuvstvitelnosti pri proektirovanii konstruktsii [Sensitivity analysis in structural design]. Moscow, Mir Publ., 1988, 428 p. (In Russ.).

**[5]**Huang Sh., Kostin V.A., Lapteva E.Yu. Vestnik Moskovskogo aviatsionnogo instituta — Aerospace MAI Journal, 2018, vol. 25, no. 3, pp. 64–72.

**[6]**Tushev O.N., Berezovskiy A.V. Vestnik MGTU im. N. E. Baumana. Ser. Mashinostroenie — Herald of the Bauman Moscow State Technical University. Series Mechanical Engineering, 2007, no. 1, pp. 35–44.

**[7]**Heylen W., Lammens S., Sas P. Modal Analysis Theory and Testing. K.U. Leuven, Belgium, 1997. [In Russ.: Heylen W., Lammens S., Sas P. Modalny analiz: teoriya i ispytaniiy. Moscow, JS Novatest Publ., 2010, 319 p.].

**[8]**Batseva O.D., Dmitriev S.N. Inzhenerny zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2018, iss. 7. http://dx.doi.org/10.18698/2308-6033-2018-7-1785.

**[9]**Gantmacher F.R. Matrix Theory. Chelsea, 1960. [In Russ.: Gantmacher F.R. Teoriya matrits. Moscow, Fizmatlit Publ., 2010, 558 p.].

**[10]**Belmman R. Introduction to Matrix Analysis. Society for Industrial and Applied Mathematics. 2nd ed., 1987, 430 p. [In Russ.: Belmman R. Vvedenie v teoriyu matrits. Moscow, Nauka Publ., 1969, 367 p.].