doi: 10.17586/2226-1494-2015-15-5-839-848


A. S. Vasilyev, A. V. Ushakov

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For citation: Vasilyev A.S., Ushakov A.V. Modeling of dynamic systems with modulation by means of Kronecker vector-matrix representation. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 5, pp. 839–848.


The paper deals with modeling of dynamic systems with modulation by the possibilities of state-space method. This method, being the basis of modern control theory, is based on the possibilities of vector-matrix formalism of linear algebra and helps to solve various problems of technical control of continuous and discrete nature invariant with respect to the dimension of their “input-output” objects. Unfortunately, it turned its back on the wide group of control systems, which hardware environment modulates signals. The marked system deficiency is partially offset by this paper, which proposes Kronecker vector-matrix representations for purposes of system representation of processes with signal modulation. The main result is vector-matrix representation of processes with modulation with no formal difference from continuous systems. It has been found that abilities of these representations could be effectively used in research of systems with modulation. Obtained model representations of processes with modulation are best adapted to the state-space method. These approaches for counting eigenvalues of Kronecker matrix summaries, that are matrix basis of model representations of processes described by Kronecker vector products, give the possibility to use modal direction in research of dynamics for systems with modulation. It is shown that the use of controllability for eigenvalues of general matrixes applied to Kronecker structures enabled to divide successfully eigenvalue spectrum into directed and not directed components. Obtained findings including design problems for models of dynamic processes with modulation based on the features of Kronecker vector and matrix structures, invariant with respect to the dimension of input-output relations, are applicable in the development of alternate current servo drives.  

Keywords: system with modulation, modeling, Kronecker vector-matrix representation, eigenvalues controllability.

Acknowledgements. The work was partially financially supported by the Government of the Russian Federation (Grant 074-U01) and by the Ministry of Education and Science of the Russian Federation (Project 14.Z50.31.0031).

1. Dudarenko N.A., Slita O.V., Ushakov A.V. Matematicheskie Osnovy Sovremennoi Teorii Upravleniya: Apparat Metoda Prostranstva Sostoyanii [Mathematical Foundations of Modern Control Theory: The Apparatus of the State Space Method] Ed. A.V. Ushakov. St. Petersburg, SPbSU ITMO Publ., 2008, 323 p.
2. Kurakin K.I., Kurakin L.K. Analiz Sistem Avtomaticheskogo Regulirovaniya na Nesushchei Peremennogo Toka [Analysis of Automatic Control Systems on the Carrier AC]. Moscow, Mashinostroenie Publ., 1978, 240p.
3. Sabinin Yu.A. Pozitsionnye i Sledyashchie Elektromekhanicheskie Sistemy: Uchebnoe posobie [Positional and Tracking Electromechanical Systems: Textbook]. St. Petersburg, Energoatomizdat Publ., 2001, 208 p.
4. Zadeh L.A., Desoer C.A. Linear System Theory: The State Space Approach. NY, McGraw-Hill Book Company Inc., 1963, 612 p.
5. Tou J.T. Modern Control Theory. NY, McGraw-Hill, 1964, 343 p.
6. Lancaster P., Tismenetskiy M. The Theory of Matrices. 2nd ed. with Applications. Ser. Computer Science and Applied Mathematics. Academic Press, 1985, 570 p
7. Gantmakher F.R. Matrix Theory. Moscow, Nauka Publ., 1973, 575 p. (In Russian)
8. Wilkinson J.H. The Algebraic Eigenvalue Problem. Oxford, 1984, 680 p.
9. Golub G.H., Van Loan C.F. Matrix Computations. Baltimore, Johns Hopkins University Press, 1996, 728 p.
10. Voevodin V.V., Kuznetsov Yu.A. Matritsy i Vychisleniya [Matrices and Calculations]. Moscow, Nauka Publ.,
1984, 320 p.
11. Godsil Ch.D., Royle G. Algebraic Graph Theory. NY, Springer-Verlag, 2001, 443 p.
12. Biryukov D.S., Ushakov A.V. Cost control estimation for harmonic exogenous actions: gramian approach. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2011, no. 2 (72), pp. 117–122.
13. Akunov T.A., Alisherov S., Omorov R.O., Ushakov A.V. Matrichnye Uravneniya v Zadachakh Upravleniya i Nablyudeniya Nepreryvnymi Ob"ektami [Matrix Equations in Control and Continuous Monitoring Objects]. Bishkek, Ilim, 1991, 174 p.
14. Bellman R. Introduction to Matrix Analysis. NY, McGraw-Hill, 1970.
15. Andreev Yu.N. Upravlenie Konechnomernymi Lineinymi Ob"ektami [Control of Finite Linear Objects]. Moscow, Nauka Publ., 1976, 424 p.
16. Besekerskii V.A., Popov E.P. Teoriya Sistem Avtomaticheskogo Regulirovaniya [The Theory of Automatic Control Systems]. St. Petersburg, Professiya Publ., 2003, 752 p.

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