Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Analytical Formula of Calculating a Controller for Linear Simo-system

Author(s):

Evgeny Anatolievich Mikrin

PSC Korolev RSC “Energia”
Russia, 141070, Moscow region, Korolev, Lenin Street, Building 4a
RAS Acad.,
Dr. Sci. (Eng.),Professor
General Designer, First Deputy General Director

Eugeny.Mikrin@rsce.ru

Nikolay Evgenievich Zubov

Bauman MSTU,
Rocket and Space Techniques faculty
Dep. of Automatic Control Systems,
Russia, 105005, Moscow, 2-nd Bauman Street, Building 5
Dr. Sci. (Eng.), Professor, Dean

Nik.Zubov@gmail.com

Alexey Vladimirovich Lapin

Bauman MSTU,
Rocket and Space Techniques faculty
Dep. of Automatic Control Systems,
Russia, 105005, Moscow, 2-nd Bauman Street, Building 5
Senior Lecturer

AlexeyPoeme@yandex.ru

Vladimir Nikolaevich Ryabchenko

JSC RDC at FGC of UES
Russia, 115201, Moscow, Kashirskoe highway, House 22, Building 3
Senior Technologist, Associate Professor
Bauman MSTU,
Rocket and Space Techniques faculty
Dep. of Automatic Control Systems,
Russia, 105005, Moscow, 2-nd Bauman Street, Building 5
Professor
Dr. Sci. (Eng.)

Ryabchenko.VN@yandex.ru

Abstract:

A compact analytical formula for calculating the coefficients of feedback (controller matrix) is obtained for linear stationary dynamic single-input multiple-output (SIMO) system while solving the problem of synthesis of linear control by fully measured state vector. This formula was obtained basing both on the technique of multilevel decomposition applied to the mathematical model of system while synthesizing its modal control (providing the desirable eigenvalues/poles placement) and on widely known property of inverse matrix of controllability in linear stationary dynamical systems. The main transformations in the mentioned technique are being performed using matrix zero divisors that are rectangular matrices zeroing their products with specified matrices. Two lemmas and the theorem for solving the assigned problem have been formulated and proved. The proposed approach allows realizing the search of scalar control bypassing intermediate calculations on each decomposition level and therefore eliminates an occurrence of badly conditioned matrices on the lower decomposition levels if the control object is described by numerical matrices. At assigned eigenvalues placement of a closed-loop SIMO-system the obtained formula matches Ackermann’s formula if the coefficients of desirable characteristic polynomial are represented by eigenvalues. Thus the calculations given in the article may be considered as an alternate technique of proving Ackermann’s formula. The efficiency of applying the described analytical formula to calculate the coefficients of feedback (controller matrix) is shown for the fourth-order control object at solving the problem of stabilizing a space vehicle longitudinal motion relative to the nominal trajectory in the Earth's atmosphere by changing the bank angle.

Keywords

References:

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