Absolute stability of multidimensional regulated systems. Aizerman problem
DOI:
https://doi.org/10.26577/JMMCS-2019-1-619Keywords:
Non-singular transformation, improper integrals, absolute stability, Aizerman problem, properties of solutionsAbstract
We consider one class of ordinary differential equations describing the dynamics of multidimensional controlled systems with a single equilibrium state with nonlinearities from a given set. Such uncertainty of the nonlinear function generates a non-uniqueness of the solution, which leads to the study of the properties of solutions of equations with differential inclusions. A new method for studying the absolute stability of the equilibrium state of controlled systems with many
continuous nonlinearities with incomplete information about them is proposed. By non-singular transformation, the original system is reduced to a special form, which allows using information about the properties of nonlinearities. We study properties of the solutions, obtain estimates for the solutions of the original system and the transformed system, and prove their boundedness. The identities with respect to the components of the nonlinear function are obtained and their
connection with the phase variables is established. Estimates of improper integrals along the solution of the system are obtained and they are used to obtain conditions for absolute stability.
The class of multidimensional nonlinear controlled systems for which the problem of Aizerman has a solution is highlighted. For this class of regulated systems, necessary and sufficient conditions for absolute stability are obtained.
References
[2] Lure A.I. Nekotoryie nelineynyie zadachi teorii avtomaticheskogo regulirovaniya [Some nonlinear problems in the theory of automatic control], (M.: Gostehizdat, 1951) : 216.
[3] Popov V.M. Giperustoychivost avtomaticheskih sistem [Hyperstability 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 non-unique equilibrium state], (M.: Nauka, 1978) : 400.
[5] Aysagaliev S.A. Ob opredelenii oblasti absolyutnoy ustoychivosti vyinuzhdennyih dvizheniy v nelineynyih sistemah [On the determination of the region of absolute stability of forced motions in nonlinear systems], (Izv. AN SSSR. Tehnicheskaya kibernetika, 1969) : 38-48.
[6] Aysagaliev S.A. Ob opredelenii oblasti absolyutnoy ustoychivosti sistemyi upravleniya s neskolkimi nelineynyimi elementami [Determining the region of absolute stability of a control system with several nonlinear elements], (AN SSSR. Avtomatika i telemehanika, 1970) : 83-94.
[7] Ayzerman M.A. Ob odnoy probleme, kasayuscheysya ustoychivosti v "bolshom"dinamicheskih sistem [On one problem concerning stability in the “large” dynamic systems], (UMN, T. 4. V 4, 1949) : 186-188.
[8] Kalman R.E. Physical and Mathematical mechanisms of instability in nonlinear automatic control systems (Transactions of ASME, Vol. 79.3., 1957) : 553-556.
[9] Pliss V.A. O probleme Aizermana dlya sluchaya sistemyi treh differentsialnyih uravneniy [On the problem of Aizerman for the case of a system of three differential equations], (Dokl. AN SSSR, 3:121, 1958).
[10] Aysagaliev S.A. K teorii absolyutnoy ustoychivosti reguliruemyih sistem [To the theory of absolute stability of regulated systems], (Differentsialnyie uravneniya, Minsk-Moskva, T. 30. V 5, 1994) : 748-757.
[11] Aisagaliev S.A., Aipanov Sh.A. K teorii globalnoy asimptoticheskoy ustoychivosti fazovyih sistem [To the theory of global asymptotic stability of phase systems], (Differentsialnyie uravneniya, Vol. 8, No 30, 1999) : 3–11.
[12] Aisagaliev S.A., Kalimoldayev M.N. Certain problems of Synchronization theory (Journal Inverse Ill Posed Problems, 2013) : 159-175.
[13] Aysagaliev S.A. Problema Aizermana v teorii absolyutnoy ustoychivosti reguliruemyih sistem [The problem of Aizerman in the theory of absolute stability of regulated systems], (Matematicheskiy sbornik, IMRAN, T. 209, V 6, 2018) : 3–24. [14] Krasovskiy N.N. Nekotoryie zadachi teorii ustoychivosti dvizheniya [Some problems of the theory of stability of motion], (M.: Fizmatgiz, 1959) : 211.
