Rate of convergence to periodic regimes in nonlinear feedback systems with strongly convex backlash characteristics

Igor G. Vladimirov; Ian R. Petersen; Igor G. Vladimirov; Ian R. Petersen

doi:10.3934/mine.2025025

Mathematics in Engineering

2025, Volume 7, Issue 5: 610-636. doi: 10.3934/mine.2025025

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Rate of convergence to periodic regimes in nonlinear feedback systems with strongly convex backlash characteristics

Igor G. Vladimirov ^,,
Ian R. Petersen

School of Engineering, Australian National University, Canberra, ACT 2601, Australia

Received: 05 January 2025 Revised: 20 August 2025 Accepted: 17 September 2025 Published: 22 September 2025

This paper considers a class of hysteresis systems consisting of a linear part with an external input and feedback with a backlash nonlinearity. Assuming that the latter is specified by a strongly convex set, we establish estimates for the Lyapunov exponents which quantify the rate of convergence of the system state trajectories to a forced periodic regime when the input is a periodic function of time with a sufficiently large "amplitude". These results employ enhanced dissipation inequalities, arising from differential inclusions with strongly convex sets which were used previously for the Moreau sweeping process.
- backlash dynamics,
- strong convexity,
- dissipation inequality,
- periodic regime,
- Lyapunov exponent
Citation: Igor G. Vladimirov, Ian R. Petersen. Rate of convergence to periodic regimes in nonlinear feedback systems with strongly convex backlash characteristics[J]. Mathematics in Engineering, 2025, 7(5): 610-636. doi: 10.3934/mine.2025025

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Abstract

This paper considers a class of hysteresis systems consisting of a linear part with an external input and feedback with a backlash nonlinearity. Assuming that the latter is specified by a strongly convex set, we establish estimates for the Lyapunov exponents which quantify the rate of convergence of the system state trajectories to a forced periodic regime when the input is a periodic function of time with a sufficiently large "amplitude". These results employ enhanced dissipation inequalities, arising from differential inclusions with strongly convex sets which were used previously for the Moreau sweeping process.

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