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

Mathematical Aspects of Condition of MIMO-system Invariance to Disturbances in Control Channels

Author(s):

Nikolay Evgenievich Zubov

Doctor of Technical Sciences, Professor, Professor of Department of Automatic Control Systems,
Dean of Rocket and Space Techniques Faculty at Bauman Moscow State Technical University (Bauman MSTU),
Professor of Postgraduate Studies at S.P. Korolev Rocket and Space Corporation "Energia" (S.P. Korolev RSC "Energia")

Nik.Zubov@gmail.com

Vladimir Nikolaevich Ryabchenko

Doctor of Technical Sciences, Associate Professor, Professor of Department of Automatic Control Systems
at Bauman Moscow State Technical University (Bauman MSTU)

RyabchenkoVN@yandex.ru

Alexey Vladimirovich Lapin

Candidate of Technical Sciences, Associate Professor of Department of Automatic Control Systems
at Bauman Moscow State Technical University (Bauman MSTU), 2nd category engineer
at State Research Institute of Aviation Systems (GosNIIAS)

AlexeyPoeme@yandex.ru

Abstract:

This paper presents constructive conditions of invariance of linear dynamic system with multi inputs and multi outputs (MIMO-system) to disturbances in control channels. The approach to invariant control synthesis consists in searching such matrix of feedback coefficients of linear system that fulfills invariance conditions represented by a system of polynomial matrix equations of a certain structure. We obtain these conditions basing on the solution of symmetric matrix equation regularization task. Theorems with proofs and illustrative examples are demonstrated for mathematic approach to analytic synthesis of invariant system with multi inputs and multi outputs (MIMO-system) as well as for numeric synthesis of a single-rotor helicopter spatial motion control system. In the numeric example the mathematic statement of control task has allowed to organize "insensitivity" of roll and pitch angles to disturbances in control channels, providing at the same time stability of general motion of the flying vehicle.

Keywords

• disturbance in control channel
• exact pole placement
• invariant control
• Morse injection
• single-rotor helicopter

References:

1. Schipanov, G. V. Teoriya i metody proektirovaniya avtomaticheskikh regulatorov [Theory and methods of designing automatic regulators]. Avtomat. I Telemekh., iss. 1, p. 49-66, 1939. (In Russian)
2. Schipanov, G. V. Giroskopicheskie pribory slepogo poleta [Gyroscopic devices for blind flight]. Moscow, Oborongiz, 1938, 116 p. (In Russian)
3. Schipanov G. V. i teoriya invariantnosti (trudy i dokumenty) [Schipanov G. V. and invariance theory (works and documents)]: Origin. Lezina, Z. M., and Lezin, V. I. Moscow, Fizmatlit, 2004
4. Uonem, M. Linejnye mnogomernye sistemy upravleniya: geometricheskii podkhod [Linear multidimensional control systems: the geometrical approach]. Moscow, Nauka Publ., 1980, 376 p. (In Russian)
5. Chen, H. L. General Decoupling Theory of Multivariable Process Control Systems. Springer-Verlag, 1983
6. Dion, J. M., and Commault C. Feedback Decoupling of Structured Systems. IEEE Trans. on Automat. Contr., vol. 38, iss. 7, p. 1132-1135, 1993. DOI: 10. 1109/9. 231471
7. Van der Woude, J. W., and Murota K. Disturbance Decoupling with Pole Placement for Structured Systems: A Graph-Theoretic Approach. SIAM J. on Matr. Anal. and Appl., vol. 16, iss. 3, p. 922-942, 1995. DOI: 10. 1137/S0895479893251344
8. Wang, Q. G. Decoupling Control. Springer-Verlag, 2003
9. Misrikhanov, M. Sh. Invariantnoe upravlenie mnogomernymi sistemami: algebraicheskii podkhod [Invariant control of multidimensional systems: the algebraic approach]. Мoscow, |Nauka Publ., 2007, 284 p. (In Russian)
10. Zubov, N. E., Mikrin, E. A., Misrikhanov, M. Sh., and Ryabchenko V. N. Invariance Conditions for MIMO-Systems Based on Regularization. Doklady Mathematics, vol. 92, iss. 3, p. 554-666, 2015. DOI: 10. 1134/S106456241506006X
11. Morse, A. S. Structural Invariants of Linear Multivariable Systems. SIAM J. Control Optim., vol. 11, iss. 3, p. 446-465, 1973. DOI: 10. 1137/0311037
12. Bukov, V. N., Goryunov, S. V., and Ryabchenko, V. N. Matrix Linear Systems: A Comparative Review of the Approachesto their Analysis and Synthesis. Autom. Remote Control, vol. 61, iss. 11, part 1, p. 1759-1795, 2000
13. Misrikhanov, M. Sh., and Ryabchenko, V. N. Pole Placement for Controlling a Large Scale Power System. Autom. Remote Control, vol. 72, iss. 10, p. 2123-2146, 2011. DOI: 10. 1134/S0005117911100110
14. Zubov, N. E., Mikrin, E. A., Oleynik, A. S., Ryabchenko, V. N., and Efanov, D. E. Otsenka uglovoy skorosti kosmicheskogo apparata v regime orbital’noy stabilizatsii po rezul’tatam izmereniy datchika mestnoy vertikali [The spacecraft angular velocity estimation in the orbital stabilization mode by the results of the local vertical sensor measurements]. Herald of Bauman MSTU, Series Instrument Engineering, iss. 5, p. 3-15, 2014. (In Russian)
15. Zubov, N. E., Mikrin, E. A., Ryabchenko V. N., and Proletarskii, A. V. Analytical Synthesis of Control Laws for Lateral Motion of Aircraft. Russian Aeronautics, vol. 58, iss. 3, p. 263-270, 2015. DOI: 10. 3103/S1068799815030034
16. 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, 666 p. (In Russian)
17. Gadjiev, M. G., Misrikhanov, M. Sh., Ryabchenko, V. N., and Sharov Yu. V. Matrichnye metody analiza I upravleniya perekhodnymi protsessami v electroenergeticheskikh sistemakh [Matrix methods of analysis and control of transients in electric power systems]. Moscow, MPEI Publ. House, 2019. (In Russian)

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