Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control

• Published in: Automatica (Oxford) Vol. 44; no. 7; pp. 1923 - 1928
• Main Authors: Liu, Z.H., Zhu, W.Q.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.07.2008; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Control theory. Systems; Exact sciences and technology; Feedback control; Fundamental areas of phenomenology (including applications); Lyapunov exponent; Nonlinear system; Physics; Solid mechanics; Stochastic averaging; Stochastic stability; Structural and continuum mechanics; System theory; Time delay; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Stochastic stability; Feedback control; Time delay; Stochastic averaging; Nonlinear system; Lyapunov exponent; Averaging method; Itô equation; Necessary and sufficient condition; Differential equation; Random excitation; Non linear control; Delay control; Feedback regulation; Integrable system; Variable delay; System with n degrees of freedom; Asymptotic stability; Linearized equation; White noise; Delay time; Stochastic equation; Parametric excitation; Probabilistic approach; Delay system; Closed feedback; Hamiltonian mechanics; Hamiltonian system; Lyapunov function
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2007.10.038

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and nonresonant Hamiltonian systems with time-delayed feedback control subject to multiplicative (parametric) excitation of Gaussian white noise is studied. First, the 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-integrable and nonresonant Hamiltonian system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi-integrable Hamiltonian 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 above mentioned procedure and its validity and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of the system.