ON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS
Анотація
A problem of description of algebraic invariants for a linear control system that
determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.
Ключові слова
Повний текст:
PDF (English)Посилання
M. Bader, Quivers, geometric invariant theory, and moduli of linear dynamical systems, Linear Algebra Appl., 428(2008), 2424 – 2454.
V. Ye. Belozyorov, Design of linear feedback for bilinear control systems, Int. J. Applied Mathematics and Computer Science, 12(4)(2002), 493 – 511.
V. Ye. Belozyorov, S. A. Volkova, Geometric approach to the problem of stabilization of control systems, Dnipro University Press, 2006 (in Russian).
V. Ye. Belozyorov, Invariant approach to an existence problem of nontrivial asymptotic stability cone, Canadien Applied Mathematics Quarterly, 15(2007), 125 – 168.
V. Ye. Belozyorov, New solution method of linear static output feedback design problem for linear control systems, Linear Algebra Appl., 504(2016), 204 – 227.
V. Ye. Belozyorov, Invariant approach to the solvability of feedback design problem for linear control systems, Modelling, Dnipro National University Press, 25(2017), 58 – 69.
V. Ye. Belozyorov, Ye. M. Kosariev, M. M. Pulin, V. G. Sychenko, V. G. Zaytsev. A new mathematical model of dynamic process in direct current traction power supply system, Journal of Optimization, Differential Equations and Their Applications (JODEA), 27(1) (2019), 21 – 55.
A. E. Browder, J. Draisma, M. Popoviciu, The degrees of a system of parameters of the ring of invariants of a binary form, Tranform. Groups, 20(2015), 953 – 967.
C. I. Byrnes, N. E. Hurt, On the moduli of linear dynamical systems, Studies in Analysis, Adv. Math. Suppl. Stud., Academic Press, New York, 4(1979), 83 – 122.
B. De Bruyn, On polyvectors of vector spaces and hyperplanes of rojective Grassmannians, Linear Algebra Appl., 435(2011), 1055 – 1084.
R. T. Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud, Neural ordinary differential equations, arXiv preprint arXiv:1806.07366v5[cs.LG], (2019), 1 – 18.
D. F. Delchamps, Global structure of families of multivariable linear systems with an application to identification, Math. Systems Theory, 18(1985), 329 – 380.
S. Friedland, Classification of linear systems, Contemporary Mathematics, 47(1985), 131 – 147.
P. A. Fuhrmann, U. Helmke, Equivalence conditions for behaviors and Kronecker canonical form, Math. Control Signals Syst, 22(2011), 267 – 293.
F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publising, Providence, Rhode Island, 2000.
N. V. Georgiev, P. N. Gospodinov, V. G. Petrov, Multi-variant time series based reconstruction of dynamical systems, Advanced Modeling and Optimization, 8 (2006), 53 – 64.
S. Haykin, Neural Networks. A Comprehensive Foundation, Second Edition, Pearson Education, Prentice Hall, 2005.
M. Hazewinkel, (Fine)moduli (spaces) for linear systems: what are they and what are they good for? Geometrical Methods for the Study of Linear Systems, (Proc. NATO Adv. Study Inst., Harvard Univ., Cambridge, Mass, 1979), NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci., Reidel, Dordrecht, 62(1980), 125 – 193.
M. Hazewinkel, C. Martin, Representations of the symmetric group, the specialization order, systems and grassmann manifolds, L’Enseingnement Mathematique, 29(1983), 53 – 87.
M. Hazewinkel, Lectures on invariants, representations and Lie algebras in systems and control theory, Lecture Notes in Mathematics, Springer- erlag, Berlin, Heidelberg, New York, 1029, 1984.
U. Helmke, The topology of a moduli space for linear dynamical systems, Comment. Math. Helvetici, 60(1985), 630 – 655.
U. Helmke, The topology of the moduli space for reachable linear dynamical systems: the complex case, Math. Systems Theory, 19(1986), 155 – 187.
U. Helmke, A compactification of the space of rational transfer functions by singular systems, J. Math. Syst. Estim.Control, 3(4)(1993), 459 – 472.
D. Hinrichsen, D. Pratzel-Wolters, State and input transformations for reachable systems: a polynomial approach, Contemporary Mathematics, 47(1985),217 – 239.
J. J. Hopfield, D.W. Tank, Computing with neural circuits: a model, Science, New Series, 233(4764)(1985), 625 – 633.
E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT Press Cambridge, Massachusetts, London, England, 2007.
J. Jia, A. R. Benson, Neural jump stochastic differential equations, arXiv preprint arXiv:1905.10403v3[cs.LG], (2020), 1 – 14.
R. E. Kalman, Algebraic geometric description of the class of linear systems of constant dimension, Proceedings of the Annual Princeton Conference on Information Sciences and Systems, 8(1974), 189 – 192.
Y.- C.Lai, N. Ye, Recent developments in chaotic time series analysis , Int. J. Bifurc. Chaos, 13 (2003), 1383 – 1422.
M. Li, Fractal time series – a tutorial review, Mathematical Problems in Engineering, 2010 (2010), 157264-1 – 26.
Z. Liu, Chaotic time series analysis, Mathematical Problems in Engineering, 2010 (2010), 720190-1 – 31.
V. G. Lomadze, Finite-dimensional time-invariant linear dynamical systems: algebraic theory, Acta Appl. Math., 19(2) (1990), 149 – 201.
W. Manthey, U. Helmke, Bruhat canonical form for linear systems, Linear Algebra Appl., 425(2007), 261 – 282.
N. Marwan, M. Romano, M. Thiel, J. Kurths, Recurrence plots for the analysis of complex systems, Physics Reports, 438 (2007), 237 – 329.
D. Mumford, J. Fogarty, Geometric Invariant Theory, Second Enlarged Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
V. G. Petrov, J. Kurths, N. V. Georgiev, Reconstructing differential equations from a time series, Int. J. Bifurc. Chaos, 13 (2003), 3307 – 3323.
C. Processi, The invariant theory of nn-matrices, Advances in Math., 19(1976), 306 – 381.
M. S. Ravi, J. Rosenthal, X. Wang, Dynamic pole assigment and Schubert calculus, SIAM J. Control and Opt., 34(3)(1996), 813 – 832.
M. S. Ravi, J. Rosenthal, U. Helmke, Output feedback invariants, Linear Algebra Appl., 351 – 352(2002), 623 – 637.
L. E. Renner, Orbits and invariants of visible group actions, Tranform. Groups, 17(2012), 1191 – 1208.
J. Rosenthal, A compactification of the space of multivariable linear systems using geometric invariant theory, J. Math. Syst. Estim. Control, 2(1)(1992), 111 – 121.
J. Rosenthal, X. Wang, Output feedback pole placement with dynamic compensator, IEEE Trans. Autom. Control, AC-41(2)(1996), 830 – 843.
V. Sychenko, V. Kuznetsov, Ye. Kosariev, P. Hubskyi, V. Belozyorov, V. Zaytsev, M. Pulin, Development of an approach to ensure stability of the traction direct current system, Eastern-European Journal of Enterprise Technologies, 95(5)(2018), 47 – 57.
T. A. Springer, Invariant Theory, Springer-Verlag, New York, 1977.
A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 845(1981), 1 – 180.
A. Tannenbaum, Invariant theory and families of dynamical systems, Mathematical System Theory – The Influence of R. E. Kalman, Springer, New York, (1991), 327 – 345.
A. Tannenbaum, On the stabilizer subgroups of a pair of matrices, Linear Algebra Appl., 50(1983), 527 – 544.
N.-E. Tatar, Hopfield neural networks with unbounded monotone activation functions, Hindawi Publishing Corporation. Advances in Artificial Neural Systems, 2012, 2012, 571358-1 – 5.
C. L. Webber, N. Marwan (eds), Reccurence Quantification Analysis. Theory and Best Practice, Springer, NY, Dordrecht, Heidelberg, London, 2015.
W. M. Wonham, Linear Multivariable Control. A Geometric Approach, Springer- Verlag, New-York, 1985.
H. Zhang, X. Gao, J. Unterman, T. Arodz, Approximation capabilities of neural ordinary differential equations, arXiv preprint arXiv: 1907.12998v1[cs.LG], (2019), 1 – 11.
DOI: http://dx.doi.org/10.15421/142002
Посилання
- Поки немає зовнішніх посилань.
Індексування журналу
Журнал розміщено у наукометричних базах, репозитаріях та пошукових системах:
Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І.
email: p.kogut@i.ua
Це видання має доступ за ліцензією Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.