Improper integrals for stability theory of multidimensional regulated systems
DOI:
https://doi.org/10.26577/jmmcs-2017-3-464Keywords:
Nonsingular transformation,, improper integrals,, absolute stability,, Aizerman problem,, absolute stability sectorsAbstract
A class of ordinary differential equations described the dynamics of nonlinear regulated systems
the right-hand part of which contains the nonlinear functions of the given set is considered. The
uncertainty of the right-hand side arises the non-uniqueness of the solution, that leads to the
necessity to study the group properties of solutions of the system. One such property is the absolute
stability of the trivial solution, i.e. properties at which all decisions coming from any starting point
for any non-linear functions of the given set tend over time to an equilibrium position. A completely
new method for the study of absolute stability of nonlinear regulated systems without involving
any Lyapunov functions and frequency theorems is proposed by evaluating improper integrals
along the solutions of the system. The motion equations of the system is led to a special form
by non-singular transformation, which allows to represent the integrand improper integrals as the
sum of two terms. The first term is a quadratic form reduced to the diagonal form, and the second
term is the total differential function on time. The representation of the integrand, ultimately,
leads to easily verifiable criteria for absolute stability.
References
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.
