Analysis and Synthesis of Dynamic MIMO-system Based on Band Matrices of Special Type
Author(s):
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
Ryabchenko.VN@yandex.ru
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
Nik.Zubov@gmail.com
Alexey Vladimirovich Lapin
Senior Lecturer of Dep. "Automatic Control Systems" at Bauman MSTU
Russia, 105005, Moscow, 2-nd Bauman Street, Building 5
AlexeyPoeme@yandex.ru
Abstract:
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.
Keywords
- 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
References:
- Andreev Yu. N., Upravleniye konechnomernymi linejnymi ob'ectami [Control of Finite-Dimensional Linear Objects]. Moscow, Nauka Publ., 1976. (In Russian)
- Voronov A. A., Vvedeniye v dinamiku slozhnykh upravlyaemykh system [Introduction to Complex Controllable Systems Dynamics]. Moscow, Nauka Publ., 1985. (In Russian)
- Dorf R. C, and Bishop R. H., Modern Control Systems. NJ, Pearson Education Inc., 2017
- Gadzhiev M. G., Misrikhanov M. Sh., Ryabchenko V. N., and Sharov Yu. V., Matrichnye metody analiza i upravleniya perekhodnymi processami v e’lektroe’nergeticheskikh sistemakh [Matrix Methods of Analysis and Control of Transients in Electric Power Systems]. Moscow, MPEI Publ. House, 2019. (In Russian)
- 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., Robastnaya ustojchivost’ i upravleniye [Robust Stability and Control]. Moscow, Nauka Publ., 2002. (In Russian)
- Uonem M., Linejnye mnogomernye sistemy upravleniya. Geometricheskij podkhod [Linear Multidimensional Control Systems. The Geometrical Approach]. Moscow, Nauka Publ., 1980. (In Russian)
- Zhou K. M., and Doyle J. C., Essentials of robust control. Prentice Hall. NJ. 1999
- Kailath T., Linear Systems. Prentice Hall. NJ. 1980
- Voevodin V. V., and Kuznecov Yu. A., Matricy i vychisleniya [Matrices and Calculations]. Moscow, Nauka Publ., 1984. (In Russian)
- Demmel J. W., Applied Numerical Linear Algebra. PA. SIAM. 1997
- Misrikhanov M. Sh., and Ryabchenko V. N., "The Band Formula for A. N. Krylov’s Problem," Automation and Remote Control, vol. 68, no. 12, pp. 2142 - 2157, 2007. DOI: 10. 1134/S0005117907120041
- Nordstrom K., and Norlander H., "On the Multi Input Pole Placement Control Problem," // Proceedings of the 36th IEEE Conference on Decision and Control, vol. 5, pp. 4288 - 4293, 1998. DOI: 10. 1109/CDC. 1997. 649511
- 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
