Dichotomy and stability of disturbed systems with periodic nonlinearities

Vera B. Smirnova, Anton V. Proskurnikov, Natalia V. Utina, Roman V. Titov

Research output: Chapter in Book/Conference proceedings/Edited volumeConference contributionScientificpeer-review

4 Citations (Scopus)
17 Downloads (Pure)


Systems that can be decomposed as feedback interconnections of stable linear blocks and periodic nonlinearities arise in many physical and engineering applications. The relevant models e.g. describe oscillations of a viscously damped pendulum, synchronization circuits (phase, frequency and delay locked loops) and networks of coupled power generators. A system with periodic nonlinearities usually has multiple equilibria (some of them being locally unstable). Many tools of classical stability and control theories fail to cope with such systems. One of the efficient methods, elaborated to deal with periodic nonlinearities, stems from the celebrated Popov method of 'integral indices', or integral quadratic constraints; this method leads, in particular, to frequency-domain criteria of the solutions' convergence, or, equivalently, global stability of the equilibria set. In this paper, we further develop Popov's method, addressing the problem of robustness of the convergence property against external disturbances that do not oscillate at infinity (allowing the system to have equilibria points). Will the forced solutions also converge to one of the equilibria points of the disturbed system? In this paper, a criterion for this type of robustness is offered.

Original languageEnglish
Title of host publicationProceedings of the 26th Mediterranean Conference on Control and Automation (MED 2018)
Place of PublicationPiscataway, NJ, USA
ISBN (Print)978-1-5386-7890-9
Publication statusPublished - 2018
EventMED 2018: 26th Mediterranean Conference on Control and Automation - Zadar, Croatia
Duration: 19 Jun 201822 Jun 2018


ConferenceMED 2018: 26th Mediterranean Conference on Control and Automation


  • Convergence
  • Frequency-domain analysis
  • Circuit stability
  • Stability criteria
  • Synchronization

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