窄带随机激励下时滞反馈控制的强非线性系统随机动力学研究
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摘要
本文研究了几类窄带随机激励下时滞反馈控制的强非线性系统的随机响应、可靠性和稳定性。先借助广义谐和函数将时滞反馈控制力近似等价为无时滞的控制力形式。再运用随机平均法,将系统的运动方程化为关于响应幅值与相位差的平均It(?)随机微分方程。在此基础上,建立并求解联合概率密度满足的Fokker-Planck-Kolmogorov(FPK)方程,得到系统的响应,据此研究时滞反馈控制对系统响应的影响;建立条件可靠性函数满足的后向Kolmogorov方程以及平均首次穿越时间满足的Pontryagin方程,分别求解这些方程,研究时滞控制对系统的可靠性函数、首次穿越时间的概率密度和平均首次穿越时间的影响;引入系统响应幅值作为范数,得到最大Lvapunov指数的近似表达式,研究时滞控制对系统概率为1渐近稳定性的影响。基于上述方法,在第二章中,研究了时滞反馈控制对谐和与白噪声联合激励下强非线性系统的随机响应、可靠性及稳定性的影响;在第三章中,研究了时滞反馈控制对谐和与宽带噪声联合激励下强非线性系统的随机响应、可靠性及稳定性的影响;在第四章中,研究了时滞反馈控制对有界噪声激励下强非线性系统的随机响应和稳定性的影响。利用Monte Carlo数字模拟对所有的理论结果进行验证,研究结果表明该方法能准确预测窄带随机激励环境下时滞对受控系统的影响。
The stochastic response,reliability and asymptotic Lyapunov stability with probability one of strongly nonlinear systems with time-delayed feedback control under several kinds of narrow-band random excitations are investigated.The time-delayed feedback control forces are approximated with the control forces without time delay by using the general harmonic functions.Then the motion equations are reduced to a set of averaged It(?)stochastic differential equations by using the stochastic averaging method.The stationary solution of the Fokker-Plank-Kolmogorov(FPK)equation associated with the averaged It(?)equations is obtained.The effects of time delay on the stochastic response of the controlled systems are discussed.By solving the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged It(?)equations,the effects of the time delay on the conditional reliability function,the conditional probability density of the first-passage time and the mean first-passage time are analyzed.Based on the averaged It(?)equations and by introducing a norm in terms of response amplitude,the Lyapunov exponent which determines the asymptotic Lyapunov stability with probability one of the controlled system is obtained.How the asymptotic Lyapunov stability with probability one of the controlled system affected by the time delay is analyzed.Based on the proposed method,in chapter 2,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and Gaussian white-noise excitations are studied. In chapter 3,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and wide-band noise excitations are studied.In chapter 4,the effects of time delay on stochastic response and stability of strongly nonlinear systems under bounded noise excitation are studied. All the theoretical results are verified by Monte Carlo digital simulations.
The stochastic response,reliability and asymptotic Lyapunov stability with probability one of strongly nonlinear systems with time-delayed feedback control under several kinds of narrow-band random excitations are investigated.The time-delayed feedback control forces are approximated with the control forces without time delay by using the general harmonic functions.Then the motion equations are reduced to a set of averaged It(?)stochastic differential equations by using the stochastic averaging method.The stationary solution of the Fokker-Plank-Kolmogorov(FPK)equation associated with the averaged It(?)equations is obtained.The effects of time delay on the stochastic response of the controlled systems are discussed.By solving the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged It(?)equations,the effects of the time delay on the conditional reliability function,the conditional probability density of the first-passage time and the mean first-passage time are analyzed.Based on the averaged It(?)equations and by introducing a norm in terms of response amplitude,the Lyapunov exponent which determines the asymptotic Lyapunov stability with probability one of the controlled system is obtained.How the asymptotic Lyapunov stability with probability one of the controlled system affected by the time delay is analyzed.Based on the proposed method,in chapter 2,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and Gaussian white-noise excitations are studied. In chapter 3,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and wide-band noise excitations are studied.In chapter 4,the effects of time delay on stochastic response and stability of strongly nonlinear systems under bounded noise excitation are studied. All the theoretical results are verified by Monte Carlo digital simulations.
引文
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