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Русская версия

**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

**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

**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

**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.)

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.

- analytical formula
- controller by fully measured state vector
- linear SIMO-system
- multilevel decomposition
- poles placement
- state space

- Kalman R. E., Falb P. L., and Arbib M. A.,
*Topics in Mathematical System Theory*. - New York: McGraw-Hill, 1969. - 358 p - Uonem M.
*Linejnye mnogomernye sistemy upravleniya. Geometricheskii podkhod*[Linear Multidimensional Control Systems. The geometrical Approach]. Moscow, Nauka Publ., 1980. 376 p. (In Russian) - Skelton R. E., Iwasaki T., and Grigoriadis K.,
*A Unified Algebraic Approach to Linear Control Design*. London, UK: Taylor& Francis, 1998. 304 p - Zhou K. M., and Doyle J. C.,
*Essentials of Robust Control*. Prentice Hall, 1999. 411 p - Skogestad S., and Postlethwaite I.,
*Multivariable Feedback Control*. John Wiley & Sons Ltd, 2005. 592 p - Kautsky J., Nichols N. K., and Van Dooren P., "Robust Pole Assignment in Linear State Feedback",
*Int. J. Control*, vol. 41, no. 5, pp. 1129 - 1155, 1985 - Dorf R. C, and Bishop R. H.,
*Modern Control Systems*. - New Jersey: Pearson Education Inc., 2017. - 1106 p - Aström K. J., and Hägglund T.,
*PID Controllers: Theory, Design and Tuning*. Research Triangle Park, USA, 1995. - 343 p - Zubov N. E., Mikrin E. A., Ryabchenko V. N.
*Matrichnye metody v teorii i praktike sistem avtomaticheskogo upravleniya letatel’nykh apparatov*[Matrix methods in theory and practice of flying vehicles automatic control systems]. Moscow, Bauman MSTU Publ., 2016. 666 p. (In Russian) - Kailath T.,
*Linear Systems*. Englewood Cliffs, NJ: Prentice Hall, 1980. 682 p - Gibbard M. J., Pourbeik P., and Vowles D. J.,
*Small-Signal Stability, Control and Dynamic Performance of Power Systems*. Univ. of Adelaide Press, 2015. - 658 p - Zubov N. E., Mikrin E. A., Misrikhanov M. Sh., and Ryabchenko V. N., "Modification of the Exact Pole Placement Method and its Application for the Control of Spacecraft Motion",
*Journal of Computer and Systems Sciences International*, vol. 52, no. 2, pp. 279 - 292, 2013 - Blumthaler I., and Oberst U., "Design, Parameterization and Pole Placement of Stabilizing Output Feedback Compensators via Injective Cogenerator Quotient Signal Modules",
*Linear Algebra Appl.*, vol. 436 (5-2), pp. 963 - 1000, 2012 - Peretz Y., "A Randomized Approximation Algorithm for the Minimal-Norm Static-Output-Feedback Problem",
*Automatica*, vol. 63, pp. 221 - 234, 2016 - Zubov N. E., Ryabchenko V. N., Mikrin E. A., and Misrikhanov M. Sh., "Output Control of the Spectrum of a Descriptor Dynamical System",
*Doklady Mathematics*, vol. 93, no. 3, pp. 259 - 261, 2016 - Zubov N. E., Zybin E. Y., Mikrin E. A., Misrikhanov M. Sh., Proletarskii A. V., and Ryabchenko V. N., "Output Control of a Spacecraft Motion Spectrum",
*Journal of Computer and Systems Sciences International*, vol. 53, no. 4, pp. 576 - 586, 2014 - Zubov N. E., Vorob’eva E. A., Mikrin E. A., Misrikhanov M. Sh., Ryabchenko V. N., and Timakov S. N., "Synthesis of Stabilizing Spacecraft Control Based on Generalized Ackermann’s Formula",
*Journal of Computer and Systems Sciences International*, vol. 50, no. 1, pp. 93 - 103, 2011 - Zubov N. E., Mikrin E. A., Misrikhanov M. Sh., and Ryabchenko V. N., "Stabilization of Coupled Motions of an Aircraft in the Pitch-Yaw Channels in the Absence of Information about the Sliding Angle: Analytical Synthesis",
*Journal of Computer and Systems Sciences International*, vol. 54, no. 1, pp. 93 - 103, 2015 - Zubov N. E., Mikrin E. A., Misrikhanov M. Sh., and Ryabchenko V. N., "Output control of the Longitudinal Motion of a Flying Vehicle",
*Journal of Computer and Systems Sciences International*, vol. 54, no. 5, pp. 825 - 837, 2015 - Zubov N. E., Lapin A. V., Mikrin E. A, and Ryabchenko V. N., "Output Control of the Spectrum of a Linear Dynamic System in Terms of the Van der Woude Method",
*Doklady Mathematics*, vol. 96, no. 2, pp. 457 - 460, 2017 - Dzhabarov M. A., Zubov N. E. [On One Approach to Solving the Problem of the Spacecraft Descent in the Earth’s Atmosphere].
*Pilotiruemye polyoty v kosmos*[Manned spaceflights], 2018, no. 2, pp. 46 - 63. (In Russian) - Ackermann J., "Der Entwurf linearer Regelungsysteme im Zustandraum",
*Regeltech, Proz. -Datenverarb.*, vol. 7, pp. 297 - 300, 1972