We studied the response of harmonically and stochastically excited strongly nonlinear oscillators with delayed feedback bang-bang control using the stochastic averaging method.First,the time-delayed feedback bang-bang control force is expressed approximately in terms of the system state variables without time delay.Then the averaged It stochastic differential equations for the system are derived using the stochastic averaging method.Finally,the response of the system is obtained by solving the Fokker-Plank-Kolmogorov (FPK) equation associated with the averaged It equations.A Duffing oscillator with time-delayed feedback bang-bang control under combined harmonic and white noise excitations is taken as an example to illus-trate the proposed method.The analytical results are confirmed by digital simulation.We found that the time delay in feedback bang-bang control will deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing oscillator.
Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present paper,the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced.The applications of the stochastic averaging method in studying the dynamics of active Brownian particles,the reaction rate theory,the dynamics of breathing and denaturation of DNA,and the Fermi resonance and its effect on the mean transition time are reviewed.
The first-passage problem of dynamical power system of a single-machine-infinite-bus (SMIB) system under random perturbations is studied.First,the stochastic averaging method for quasi non-integrable generalized Hamiltonian systems is applied to reduce the equations of the SMIB system under random perturbations to a set of averaged It equations.Then,the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established and solved numerically,respectively.Finally,the proposed method is verified by using the Monte Carlo simulation of the original system.
CHEN LinCong1 & ZHU WeiQiu2 1 College of Civil Engineering,Huaqiao University,Xiamen 361021,China
