Stochastic Response, Asymptotic Stability With Probability 1 And First-passage Time Of Quasi Linear System With Multi-time-delayed Feedback Control And Wide-band Random Excitation

In the present dissertation the theory of stochastically excited and dissipated Hamiltonian systems proposed by W.Q. Zhu is generalized to study the stochastic response, the stochastic stability and the reliability of quasi-linear system under multi-time-delayed feedback control and wide-band random excitations. In the study of the stochastic response, the system equations are transformed into differential equations without time delay and the averaged Ito stochastic differential equations for the slowly varying processes are derived. The stationary solution of the averaged FPK equation associated with the averaged It(o|^) equations is obtained and the effect of time-delayed feedback control on the responses is stuied. In the study of the asymptotic Lyapunov stability with probability 1, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. By introducing a new norm, the approximate formula for the largest Lyapunov exponent is derived. The necessary and sufficient condition for the asympototic Lyapunov stability with probability 1 is obtained. In the study of the first-passage failure, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. The conditional reliability function and moments of first-passage time are obtained from solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. In addition, a time-delayed feedback control problem of partially observable linear building structure under horizontal ground acceleration excitation is formulated and converted into that of completely observable linear structure by using the separation principle. The time-delayed control forces are approximately expressed in terms of control forces without time delay. The control system is then governed by Ito stochastic differential equations for the conditional means of system states and then transformed into those for the conditional means of modal energies by using the stochastic averaging method for quasi Hamiltonian systems. The control law is assumed to be modal velocity feedback control with time delay and the unknown control gains are determined by the modal performance indices. The comparising all the theoretic results and those from Monte-Carlo simulation shows that the two results in good agreement. Furthermore, it is shown that the time delay in feedback control affect the response, the asymptotic stability with probability 1 and the first-passage time remarkably. However, the deteriotation effect can be almost eliminated if the delay time is set correctly.