Method for the transformation of complex automatic control systems to integrable form

A. M. Kamachkin ; D. K. Potapov ; V. V. Yevstafyeva

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 17 (2021) no. 2, pp. 196-212

Résumé

The article considers a class of automatic control systems that is described by a multi-dimensional system of ordinary differential equations. The right hand-side of the system additively contains a linear part and the product of a control matrix by a vector that is the sum of a control vector and an external perturbation vector. The control vector is defined by a nonlinear function dependent on the product of a feedback matrix by a vector of current coordinates. The authors solve the problem of constructing a matrix of a nonsingular transformation, which leads the matrix of the linear part of the system to the Jordan normal form or the first natural normal form. The variables included in this transformation allow us to vary the system settings, which are the parameters of both the control matrix and the feedback matrix, as well as to convert the system to an integrable form. Integrable form is understood as a form in which the system can be integrated in a final form or reduced to a set of subsystems of lower orders. Furthermore, the sum of the subsystem orders is equal to the order of the original system. In the article, particular attention is paid to cases when the matrix of the linear part has complex conjugate eigenvalues, including multiple ones.

Détail

Keywords: automatic control system, multidimensional nonlinear dynamic system, nonsingular transformation, first natural normal matrix form, a system's integrable form.
Mots-clés : Jordan's normal matrix form

Bibliographie

[1] Lur'e A. I., Certain nonlinear tasks of the automatic control theory, Techn.-theor. lit. Publ., M., 1951, 216 pp. (In Russian) | MR

[2] Troitskij V. A., “About canonical transformations of equations of the automatic control theory in the presence of the multiple roots”, Applied Mathematics and Mechanics, 21:4 (1957), 574–577 (In Russian)

[3] Letov A. M., Stability of nonlinear control systems, 2nd ed., Phys.-math. lit. Publ., M., 1962, 483 pp. (In Russian) | MR

[4] Petrov V. V., Gordeev A. A., Nonlinear servomechanisms, Machinostroenie Publ, M., 1979, 472 pp. (In Russian)

[5] Nelepin R. A., Exact analytical methods in the theory of nonlinear automatic systems, Sudostroenie Publ, Leningrad, 1967, 447 pp. (In Russian)

[6] Nelepin R. A., Kamachkin A. M., Turkin I. I., Shamberov V. N., Algorithmic synthesis of the nonlinear control systems, Leningrad University Press, L., 1990, 240 pp. (In Russian)

[7] DeRusso P. M., Roy R. J., Close C. M., Desrochers A. A., State variables for engineers, 2nd ed., Wiley-Interscience, New York, 1998, 575 pp. | MR

[8] Bohr H., “Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen”, Acta Math., 45 (1925), 29–127 | DOI | MR

[9] Astrom K. J., “Oscillations in systems with relay feedback”, Adaptive Control, Filtering and Signal Processing, Springer-Verlag, New York, 1995, 1–25 | MR | Zbl

[10] Andronov A. A., Vitt A. A., Khaikin S. E., Theory of oscillators, Physmatgiz Publ, M., 1959, 915 pp. (In Russian)

[11] Kamachkin A. M., Chitrov G. M., Shamberov V. N., “Algebraical aspects of parametrical decomposition method”, 2015 International Conference “Stability and Control Prosesses” in memory of V. I. Zubov, SCP-2015, St. Petersburg State University Press, St. Petersburg, 2015, 52–54

[12] Kamachkin A. M., Shamberov V. N., Chitrov G. M., “Special matrix transformations of essential nonlinear control systems”, 2017 International Conference “Constructive Nonsmooth Analysis and Related Topics” dedicated to the memory of V. F. Demyanov, CNSA-2017, St. Petersburg State University Press, St. Petersburg, 2017, 1–3

[13] Kamachkin A. M., Chitrov G. M., Shamberov V. N., “Normal matrix forms to decomposition and control problems for manydimensional systems”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:4 (2017), 417–430 (In Russian) | MR

[14] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Solution to second-order differential equations with discontinuous right-hand side”, Electron. Journal Differential Equations, 221 (2014), 1–6 | MR

[15] Evstaf'eva V. V., “On existence conditions for a two-point oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix”, Automat. Remote Control, 2015, no. 6, 42–56 (In Russian)

[16] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Non-existence of periodic solutions to non-autonomous second-order differential equation with discontinuous nonlinearity”, Electron. Journal Differential Equations, 2016, no. 04, 1–8 | MR

[17] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Existence of solutions for second-order differential equations with discontinuous right-hand side”, Electron. Journal Differential Equations, 2016, no. 124, 1–9 | MR

[18] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence”, International Journal Robust Nonlinear Control, 27:2 (2017), 204–211 | DOI | MR | Zbl

[19] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence”, Electron. Journal Differential Equations, 2017, no. 140, 1–10 | MR

[20] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity”, Journal Dynamical Control Systems, 23:4 (2017), 825–837 | DOI | MR | Zbl

[21] Evstaf'eva V. V., “Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero”, Ukrainian Mathematical Journal, 70:8 (2018), 1085–1096 (In Russian) | MR

[22] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Existence of periodic modes in automatic control system with a three-position relay”, International Journal Control, 93:4 (2020), 763–770 | DOI | MR | Zbl

[23] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Dynamics and synchronization in feedback cyclic structures with hysteresis oscillators”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 16:2 (2020), 186–199 (In Russian) | DOI | MR

[24] Evstaf'eva V. V., “On the existence of two-point oscillatory solutions of a perturbed relay system with hysteresis”, Differential Equations, 57:2 (2021), 169–178 (In Russian)

[25] Lankaster P., Theory of matrices, Nauka Publ, M., 1973, 280 pp. (In Russian) | MR

[26] Burns R. S., Advanced control engineering, Butterworth-Heinemann, Oxford, 2001, 464 pp.

[27] Paraskevopoulos P. N., Modern control engineering, Marcel Dekker Inc, New York, 2002, 736 pp.