(Differencialnie Uravnenia i Protsesy Upravlenia)

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Русская версия

**Evgeny Anatolievich Mikrin**

RAS Acad., Dr. Sci. (Eng.), Professor

General Designer

First Deputy General Director at PSC Korolev RSC "Energia"

141070, Moscow region, Korolev, Lenin Street, Building 4a)

**Vladimir Nikolaevich Ryabchenko**

Dr. Sci. (Phys.-Math.), Associate Professor

Senior Technologist of JSC "RDC at FGC of UES"

115201, Moscow, Kashirskoe highway, House 22, Building 3

Professor of Dep. "Automatic Control Systems" at Bauman MSTU

Russia, 105005, Moscow, 2-nd Bauman Street, House 5

**Nikolay Evgenievich Zubov**

Dr. Sci. (Eng.), Professor

Professor of Dep. "Automatic Control Systems",

Dean of "Rocket and Space Techniques" faculty at Bauman MSTU

105005, Moscow, 2-nd Bauman Street, Building 5

**Alexey Vladimirovich Lapin**

Senior Lecturer of Dep. "Automatic Control Systems" at Bauman MSTU

Russia, 105005, Moscow, 2-nd Bauman Street, Building 5

The problem of analysis and synthesis of linear controllable dynamic MIMO-systems (systems with multiple input and multiple output) using band matrices of special type is considered. The fundamental basis of suggesting approach is A.N. Krylov transformations (Krylov subspaces). The main matrix transformations applying for getting solutions are left and right zero divisors. Band matrices of special type with properties that uniquely define the property of full controllability are formed basing on mentioned transformations for linear fully controllable MIMO-system. Besides, these matrices allow analytic connecting parameters of controllable MIMO-system and coefficients of its characteristic polynomial. Obtaining the formula of this connection is founded on the well-known relationship between MIMO-system controllability matrix and the companion (canonical) Frobenius form for its characteristic polynomial. Using the obtained formula a controller is synthesized with feedback providing coefficients of characteristic polynomial of the closed-loop controlled MIMO-system matching the assigned coefficients. In simplified form (for single input systems) the formula of controller is similar to the well-known Bass – Gura and Ackermann formulas. The condition is obtained for parameterizing the set of controllers that provide the assigned characteristic polynomial of closed-loop MIMO-system and that are generated by left zero divisor of a band matrix of special type.

- band matrix of special type
- characteristic polynomial
- companion form
- control by state
- controllability
- Kronecker product
- Krylov method
- linear MIMO-system
- parameterization of set of solutions
- zero divisors

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- Zubov N. E., Mikrin E. A., and 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. (In Russian) - Polyak B. T., and Shherbakov P. S.,
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- 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., 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. DOI: 10. 1134/S1064230715010153 - 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. DOI: 10. 1134/S1064230715040140 - Zubov N. E., Mikrin E. A., Ryabchenko V. N., and Fomichev A. V., "Synthesis of Control Laws for Aircraft Lateral Motion at the Lack of Data on the Slip Angle: Analytical Solution,"
*Russian Aeronautics*, vol. 60, no. 1, pp. 64 - 73, 2017. DOI: 10. 3103/S106879981701010X