QUASI-ANALYTIC METHOD OF LINEAR DYNAMIC SYSTEMS INVERSION
DOI:
https://doi.org/10.20998/2079-0023.2022.02.07Keywords:
dynamical systems, linear differential equations, triangular block matrices, inversion problem, state space, matrix equationsAbstract
The problem of inversion of dynamic systems has become widespread while solving the problems of control, identification, and measurement problems arising during the design and research of electrical and mechanical dynamic systems. Inverting is an effective way of implementing disturbance control processes, as well as in combined control systems with a predictive model. The analysis of information sources showed that in the practical solution of most inversion problems, a number of difficulties arise, which are associated with the high sensitivity of the results in relation to the accuracy of the parameters of the mathematical model of the control object, the instability of the inverse model of non-minimum-phase objects, and the violation of the conditions of physical feasibility. The work offers an effective method of inverting linear stationary dynamic systems, free from the mentioned shortcomings in many respects. The basis of the method is the presentation of input and output signals in the form of infinite linear combinations of their derivatives. A method of determining the sequence of matrix coefficients of linear representations of input and output signals is proposed. The main theoretical result is obtaining relationships between matrix coefficients of input and output signals. The work considers mathematical models of linear dynamic systems in the form of differential equations in the state space and in the equivalent "input-output" form. The considered systems must meet the conditions of asymptotic stability, as well as the condition of equal dimensions of the input and output vectors. Requirements for mathematical models of input and output signals are given, the fulfillment of which allows, instead of infinite sums representing signals, to be limited to a finite number of terms.
References
Sain M. K, Massey J. L. Invertibility of linear time-invariant dynamical systems. IEEE Trans. Automatic Control. 1969, vol. AS– 14, № 2, рр. 141–149.
Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Automatic Control. 1969, vol. AS–14, № 3, pp. 270–276.
Ilyin A. V., Korovin S. K., Fomichev V. V. Metody robastnogo obrashcheniya dinamicheskikh sistem [Methods for Robust Inversion of Dynamical Systems]. Moscow, FIZMATLIT Publ., 2009. 219 p.
Kostenko Yu. T., Lyubchik L. M. Sistemy upravleniya s dinamicheskimi modelyami [Control systems with dynamic models]. Kharkov, Osnova Publ., 1996. 212 p.
Borukhov V. T. Kriterii obratimosti lineynykh statsionarnykh mnogomernykh sistem [Criteria for the reversibility of linear stationary multidimensional systems]. Avtomatika i telemekhanika. Kiev, 1978, is. 11, pp. 5–11.
Pukhov G. E., Zhuk K. D. Sintez mnogosvyaznykh sistem upravleniya po metodu obratnykh operatorov [Synthesis of multiply connected control systems by the method of inverse operators]. Kiev, Naukova dumka Publ., 1966. 218 p.
Krutko P. D. Obratnye zadachi dinamiki upravlyaemykh sistem. Lineynye modeli [Inverse problems of the dynamics of controlled systems. Linear Models]. Moscow, Nauka Publ., 1987. 304 p.
Willsky A. S. On the invertibility of linear systems. IEEE Tr. Aut. Control. 1974, vol. 19, pp. 272–274.
Nikolsky M. S. Ob ideal'no nablyudaemykh sistemakh [On ideally observable systems]. Differentsial'nye uravneniya. 1971, vol. 7, no 4, pp. 631–638.
Krasovsky N. N. Teoriya upravleniya dvizheniem [Motion Control Theory]. Moscow, Nauka Publ., 1968. 476 p.
Kurzhansky A. B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Management and supervision in conditions of uncertainty]. Moscow, Nauka Publ., 1977. 392 p.
Gusev M. I., Kurzhansky A. B. Obratnye zadachi dinamiki upravlyaemykh sistem [Inverse problems of dynamics of controlled systems]. Mekhanika i nauchno-tekhnicheskiy progress, vol. 1: Obshchaya i prikladnaya mekhanika. Moscow, Nauka Publ., 1987, pp. 187–195.
Anikin S. A., Gusev M. I. Otsenivanie vozmushchayushchikh sil po izmereniyam parametrov dvizheniya [Estimation of Disturbing Forces from Measurements of Motion Parameters]. Garantirovannoe otsenivanie i zadachi upravleniya. Sverdlovsk, UNTs AN SSSR Publ., 1986. pp. 19–30.
Anikin S. A. Ob otsenke pogreshnosti metoda regulyarizatsii A. N. Tikhonova v zadachakh vosstanovleniya vkhodov dinamicheskikh sistem [On the estimation of the error of A. N. Tikhonov's regularization method in problems of restoring the inputs of dynamical systems]. Zhurnal vychislitel'noy matematiki i mat. fiziki. 1997, no 9, pp. 1056–1067.
Granovsky V. A. Dinamicheskie izmereniya [Dynamic measurements]. Leningrad, Energoizdat Publ., 1984. 224 p.
Shestakov A. L., Iosifov D. Yu. Reshenie obratnoy zadachi dinamiki na osnove teorii modal'nogo upravleniya s ispol'zovaniem izmeryaemogo vektora parametrov sostoyaniya pervichnogo izmeritel'nogo preobrazovatelya [Solution of the inverse problem of dynamics based on the theory of modal control using the measured vector of state parameters of the primary measuring transducer]. Izvestiya Chelyabinskogo nauchnogo tsentra. 2005, no 4(30), pp. 144–149.
Shestakov A. L., Sviridyuk G. A., Zakharova E. V. Dinamicheskie izmereniya kak zadacha optimal'nogo upravleniya [Dynamic Measurements as an Optimal Control Problem]. Obozrenie prikladnoy i promyshlennoy matematiki. 2009, vol. 16, no 4, pp. 732– 733.
Kutsenko O. S., Kovalenko S. V. [Dynamic measurements as a problem of inversion of controlled systems]. Metrologiya ta vymiryuval'na texnika (Metrologiya–2020). XII Mizhnarodna naukovo–texnichna konferenciya (6–8 zhovtnya 2020 r.). Zbirnyk dopovidej [Metrology and measuring technology (Metrology-2020). XII International Scientific and Technical Conference (October 6–8, 2020). A collection of reports]. Kharkiv, NNTs "Іnstitut metrologії" Publ., 2020, pp. 87–91.
Kutsenko A., Kovalenko S., Tovazhnyanskyy V. Inversion of dynamic systems for certain classes of signals. CEUR Workshop Proceedings. 2019, vol. 2353, pp. 391–401.
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