A4 Article in conference proceedings
Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture (2019)

Kuznetsov, N. V.; Kuznetsova, O. A.; Mokaev, T. N.; Mokaev, R. N.; Yuldashev, M. V.; Yuldashev, R. V. (2019). Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture. In Jadachowski, Lukaz (Eds.) 11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019, IFAC-PapersOnLine. IFAC 52 (16), 7-12. DOI: 10.1016/j.ifacol.2019.11.747

JYU authors or editors

Publication details

All authors or editors: Kuznetsov, N. V.; Kuznetsova, O. A.; Mokaev, T. N.; Mokaev, R. N.; Yuldashev, M. V.; Yuldashev, R. V.

Parent publication: 11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019

Parent publication editors: Jadachowski, Lukaz

Conference:

Place and date of conference: Vienna, Austria, 4.-6.9.2019

Journal or series: IFAC-PapersOnLine

eISSN: 2405-8963

Publication year: 2019

Volume: 52

Issue number: 16

Pages range: 7-12

Number of pages in the book: 842

Publisher: IFAC; Elsevier

Publication country: United Kingdom

Publication language: English

DOI: https://doi.org/10.1016/j.ifacol.2019.11.747

Open Access: Other way freely accessible online

Abstract

The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were constructed in such a way that one locally stable limit cycle (hidden oscillation) co-exists with a locally stable equilibrium. In this paper we demonstrate a multistable configuration of three co-existing hidden oscillations (limit cycles) and a locally stable equilibrium in the phase space of the fourth-order system, which provides a new class of counterexamples to the Kalman conjecture.

Keywords: control theory; oscillations

Free keywords: global stability; hidden attractors; multistability; Kalman conjecture; periodic oscillations

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2019

JUFO rating: 1