Investigation on absolute stability of multidimensional regulated systems. Aizerman problem

Authors

  • S. А. Aisagaliev al-Farabi Kazakh National University
  • A. M. Ayazbayeva al-Farabi Kazakh National University
        51 26

Keywords:

Absolute stability, nonsingular transformation, properties of the solutions, improper integrals, Aizerman problem, absolute stability sectors

Abstract

with many nonlinearities is supposed. The motion equation of the regular system is reduced
to a special kind by nonsingular transformation which allows to present the nonlinearities as
functions of phase variables. Estimations of the phase variables and identities along solution of
the system are obtained for systems with limited resources. Estimations of the improper integrals
are found and conditions of the absolute stability in the space of constructive parameters of
the system are formulated. Possibility of the sector existence is considered, where Aizerman
problem has solution for regular systems with limited resources. It should be noted, that frequency
condition of the V.M.Popov absolute stability for systems with many nonlinearities has not any
geometrical interpretation, as in the one-dimensional case, and its verify is difficult problem.
Therefore development of the new method for investigation on absolute stability of regular systems
is topical. Distinctive feature of the supposed method of investigation on absolute stability from
known methods is that the conditions of the absolute stability are obtained without using Lyapunov
function and frequency theorem of V.A. Yakubovich.

References

[1] Aizerman M. A., Gantmaher F. R. Absolyutnaya ustoychivost reguliruemyih sistem [Absolute stability of regulated systems], (Izdatelstvo AN SSSR, 1963) : 240.
[2] Lurie A. I. Nekotoryie nelineynyie zadachi teorii avtomaticheskogo regulirovaniya [Some nonlinear problems of automatic control theory], (M.: Gostehizdat, 1951) : 216.
[3] Popov V. M. Giperustoychivost avtomaticheskih sistem [Hyper-stability of automatic systems], (M.: Nauka, 1970) : 453.
[4] Gelig A. H., Leonov G. A., Yakubovich V. A. Ustoychivost nelineynyih sistem s needinstvennyim sostoyaniem ravnovesiya [Stability of nonlinear systems with a nonunique equilibrium state], (M.:Nauka, 1978) : 400.
[5] Aisagaliev S. A., «Ob opredelenii oblasti absolyutnoy ustoychivosti vyinuzhdennyih dvizheniy v nelineynyih sistemah» [On the determination of the domain of absolute stability forced motions in nonlinear systems], Izv. AN SSSR. Tehnicheskaya kibernetika (1969) : 38–48.
[6] Aisagaliev S. A., «Ob opredelenii oblasti absolyutnoy ustoychivosti sistemyi upravleniya s neskolkimi nelineynyimi
elementami» [On the determination of the domain of absolute stability of a control system with several nonlinear elements], AN SSSR. Avtomatika i telemehanika (1970) : 83–94.
[7] Aizerman M. A., «Ob odnoy probleme, kasayuscheysya ustoychivosti v "bolshom"dinamicheskih sistem» [On one problem concerning stability in "large"dynamical systems], UMN (1949) : 186–188.
[8] Kalman R. E., «Physical and Mathematical mechanisms of instability in nonlinear automatic control systems»,
Transactions of ASME (1957) : 553–556.
[9] Bragin V. O., Vagaytsev V. I., Kuznetsov N. V., Leonov G. A., «Algoritmyi poiska skryityih kolebaniy v nelineynyih
sistemah. Problemyi Ayzermana, Kalmana i tsepi ChUA» [Algorithms for searching hidden oscillations in nonlinear systems. The problems of Aizerman, Kalman, and ChUA chain], Izvestiya RAN. Teoriya i sistemyi upravleniya (2011) :
3–36.
[10] Aisagaliev S. A., «K teorii absolyutnoy ustoychivosti reguliruemyih sistem» [For the theory of absolute stability of regulated systems], Differentsialnyie uravneniya. Minsk-Moskva, Vol. 30. No 5 (1994) : 748–757.
[11] Aisagaliev S. A. Teoriya reguliruemyih sistem [Theory of regulated systems] (Kazakh universiteti, 2000), 234.
[12] Aisagaliev S. A. Teoriya ustoychivosti dinamicheskih sistem [Stability theory of dynamical systems] (Kazakh universiteti, 2012), 216.
[13] Aisagaliev S. A., Kalimoldayev M. N., «Certain problems of Synchronization theory», Journal Inverse Ill Posed Problems, No 21 (2013) : 159–175.
[14] Aisagaliev S. A., Ayazbayeva A. M., «Nesobstvennyie integralyi v teorii ustoychivosti mnogomernyih reguliruemyih sistem [Improper integrals for stability theory of multidimensional regulated systems]», Vestnik KazNU, ser. meh., mat., inf. (2017) : 3-20.

Downloads

How to Cite

Aisagaliev S. А., & Ayazbayeva, A. M. (2018). Investigation on absolute stability of multidimensional regulated systems. Aizerman problem. Journal of Mathematics, Mechanics and Computer Science, 96(4), 3–22. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/563