Asymptotic Lyapunov stability with probability one of quasi linear systems subject to multi-time-delayed feedback control and wide-band parametric random excitation

• Published in: Archive of applied mechanics (1991) Vol. 79; no. 11; pp. 1051 - 1061
• Main Authors: Li, Xue Ping, Liu, Zhong Hua, Huan, Rong Hua, Zhu, Wei Qiu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.11.2009
• Subjects: Asymptotic properties; Classical Mechanics; Engineering; Feedback control; Linear systems; Lyapunov exponents; Mathematical analysis; Original; Stability; Stochasticity; Theoretical and Applied Mechanics; Time delay; Stochastic averaging; Wide-band random excitation; Multi-time-delayed feedback; Lyapunov exponent
• DOI: 10.1007/s00419-008-0273-y
• ISSN: 0939-1533

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi linear systems subject to multi-time-delayed feedback control and multiplicative (parametric) excitation of wide-band random process is studied. First, the multi-time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi linear system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi linear systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the application and validity of the proposed procedure and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of system.