Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

• Published in: Acta mechanica Vol. 208; no. 1-2; pp. 55 - 62
• Main Authors: Feng, Changshui, Zhu, Weiqiu
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.11.2009; Springer; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Control; Differential equations; Dynamical Systems; Engineering; Engineering Thermodynamics; Exact sciences and technology; Feedback; Fundamental areas of phenomenology (including applications); Heat and Mass Transfer; Mathematics; Mechanical engineering; Numerical analysis; Numerical analysis. Scientific computation; Physics; Sciences and techniques of general use; Simulation; Solid Mechanics; Stochastic models; Structural and continuum mechanics; Theoretical and Applied Mechanics; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Stochastic Average Method; Delay Feedback Control; Lyapunov Exponent; Digital Simulation; Large Lyapunov Exponent; Parametric excitation; Averaging method; Itô equation; Probabilistic approach; Random excitation; Lyapunov method; Feedback regulation; Delay system; Noise control; State variable; Asymptotic stability; Non linear oscillator; Active system; Delay time; Duffing equation; Force control; Delay effect; Random vibration; Lyapunov function
• ISSN: 0001-5970; 1619-6937
• DOI: 10.1007/s00707-008-0126-3

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.