«Automation and Informatization of the fuel and energy complex»

Semi-invariants, invariants and null-forms in the space of linear control dynamical systems

UDC: 681.5:622.276+622.279
DOI: -

Authors:

¹ National University of Oil and Gas "Gubkin University", Moscow, Russia

Keywords: invariant, null-form, null-cone, dynamic control system, Hilbert–Mumford criterion, nilpotent matrix, reachability matrix, one-parameter subgroup, orbit, stability, nice multi-index selection

Annotation:

The authors of the article consider the action of similarity of a SLn(C) special linear group on the space of Σ(n,m) linear dynamical controlled systems with n-dimensional states and m-dimensional inputs determined by the replacement of coordinates in the system's state space. Linear dynamical systems are adequate mathematical models for a wide range of real objects and processes. For the specified action, the null forms of the object, introduced by D. Hilbert, are calculated by two ways in a common case and representing points in space in which a group acts and orbit closures of which contain zero, that is, points in which all non-constant homogeneous invariants of the action vanish. The first method uses the Hilbert–Mumford criterion, which states that a point is a null-form if and only if there is a one-parameter subgroup of SLn(C) the orbit closure of which contains zero. The second method is based on the results of C. Procesi about the generators of the algebra of invariants of finite families of n×n matrices, n-dimensional vectors and n-dimensional co-vectors. The main result, obtained by the authors of the article states that (A, B) linear system is null-form if and only if it is not completely reachable and A matrix is nilpotent, that is An = 0. Consequently, a finite set of equations defining a null-cone (the set of all null forms) in Σ(n,m) space is presented. The use of nice multi-indexes introduced by R. Kalman makes it possible to reduce the number of these equations. The common formula for a number of nice multi-index selections in case of n and m arbitrary natural is obtained and some examples are analyzed.

Bibliography:

1. Ermolaev A.I., Shafikov M.T. Identifikatsiya parametrov modeley fil'tratsii dlya zhestkogo vodonapornogo rezhima s tsel'yu prognozirovaniya dinamiki pokazateley razrabotki // Avtomatizatsiya, telemekhanizatsiya i svyaz' v neftyanoy prom-sti. – 1995. – № 9. – S. 13–14.
2. Ostashkov V.N. Praktikum po resheniyu inzhenernykh zadach matematicheskimi metodami: ucheb. posobie. – M.: BINOM. Lab. znaniy, 2013. – 200 s.
3. Ermolaev A.I., Trubacheva I.A., Nekrasov A.A. Algoritm optimizatsii debitov gazokondensatnykh skvazhin // Nauka i tekhnika v gazovoy prom-sti. – 2019. – № 3(79). – S. 26–34.
4. Ermolaev A.I., Skryabina A.S., Fayzrakhmanov G.G. Modelirovanie protsessov sozdaniya i ekspluatatsii vremennogo khraneniya gelievogo kontsentrata v produktivnykh gorizontakh Adnikanskogo mestorozhdeniya // Gazovaya prom-st'. – 2020. – № S4(808). – S. 64–71.
5. Direktor S., Rorer R. Vvedenie v teoriyu sistem: per. s angl. – M.: Mir, 1974. – 464 s.
6. Shumafov M.M. Stabilizatsiya lineynykh sistem upravleniya. Problema naznacheniya polyusov. Obzor // Vestn. S.-Peterb. un-ta. Matematika. Mekhanika. Astronomiya. – 2019. – T. 6, № 4. – S. 564–591.
7. Osetinskiy N.I. Obzor nekotorykh rezul'tatov i metodov v sovremennoy teorii lineynykh sistem // Teoriya sistem. Mat. metody i modelirovanie. – M.: Mir, 1989. – S. 328–379.
8. Eilenberg S. Automata, languages and machines. Part A. – New York: Academic Press, 1974. – 451 p.
9. Tannenbaum A. Invariance and system theory: Algebraic and geometric aspects. – Heidelberg: Springer-Verlag Berlin, 1981. – XII, 164 p. – DOI: 10.1007/BFb0093318
10. Andreev Yu.N. Algebraicheskie metody prostranstva sostoyaniy v teorii upravleniya lineynymi ob"ektami (obzor inostrannoy literatury) // Avtomatika i telemekhanika. – 1977. – № 3. – S. 5–50.
11. Sontag E.D. Mathematical control theory: Deterministic Finite-Dimensional Systems. – 2nd Edition. – New York: Springer-Verlag, 1998. – XVI, 532 p. – DOI: 10.1007/978-1-4612-0577-7
12. Hilbert D. Über die vollen Invariantensysteme // Mathematische Annalen. – 1893. – Vol. 42. – P. 313–373. – DOI: 10.1007/BF01444162
13. Vinberg E.B., Popov V.L. Teoriya invariantov // Algebraicheskaya geometriya – 4 (Itogi nauki i tekhn. Sovrem. problemy matematiki. Fundam. napravleniya. T. 55). – M.: VINITI, 1989. – S. 137–309.
14. Kraft Kh. Geometricheskie metody v teorii invariantov: per. s nem. – Novokuznetsk: NFMI, 2000. – 308 s.
15. Procesi C. The Invariant theory of n×n matrices // Advances in mathematics. – 1976. – Vol. 19. – P. 306–381. – DOI: 10.1016/0001-8708(76)90027-x
16. Kalman R.E., Falb P., Arbib M. Ocherki po matematicheskoy teorii sistem: per. s angl. – M.: Mir, 1971. – 400 s.
17. Uonem U.M. Lineynye mnogomernye sistemy upravleniya. Geometricheskiy podkhod: per. s angl. – M.: Nauka, 1980. – 376 s.
18. Le Bruyn L. Noncommutative geometry @n. – Antwerpen, Belgium, 2002. – VIII, 351 p.
19. Parshin A.N. David Gil'bert i teoriya invariantov // Istoriko-matematicheskie issled. Vyp. XX. – M.: Nauka, 1975. – S. 171–197.
20. Hesselink W.H. Singularities in the nilpotent scheme of a classical group // Transactions of the American Mathematical Society. – 1976. – Vol. 222. – P. 1–32. – DOI: 10.1090/S0002-9947-1976-0429875-8