非线性随机动力系统的最优多项式控制
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摘要
根据物理随机最优控制理论,发展了适用于一般非线性随机动力系统的最优多项式控制策略,考察了随机地震动作用下不同非线性水平Duffing系统的随机最优控制。结果表明,受控后系统反应的离散性大大降低、系统性态显著改善;采用能量均衡的超越概率准则,1阶线性控制器可以达到高阶非线性控制器的控制效果,这对于非线性控制器可能导致系统不稳定的场合具有重要意义。此外,相比较基于统计线性化的LQG控制,发展的非线性随机最优控制策略对Duffing系统的非线性水平不敏感、具有良好的鲁棒性,能实现系统的精细化控制,而采用名义白噪声输入的LQG控制则不具备这一特点。
According to the principle of physical stochastic optimal control,a novel stochastic optimal control strategy is developed in the context of optimal polynomial control that is well-adapted to nonlinear stochastic dynamical systems.Duffing oscillators with varying levels of nonlinearity,subjected to random ground motions,are investigated for illustrative purposes.Numerical results indicate that the statistical variability of system responses gains an obvious reduction,and the system performance improves significantly after the optimal polynomial controller is mounted.Using the control criterion with the minimum of the exceedance probability of system quantities in energy trade-off sense,the 1st-order controller suffices even for strongly nonlinear systems.This bypasses the need to utilize nonlinear controllers which may be associated with dynamical instabilities due to time delay and computational dynamics.Besides,the optimal polynomial controller exhibits high robustness to the levels of nonlinearity of Duffing oscillators.The classical statistical linearization based LQG,however,using the nominal Gaussian white noise as the external excitation does not have this advantage.
According to the principle of physical stochastic optimal control,a novel stochastic optimal control strategy is developed in the context of optimal polynomial control that is well-adapted to nonlinear stochastic dynamical systems.Duffing oscillators with varying levels of nonlinearity,subjected to random ground motions,are investigated for illustrative purposes.Numerical results indicate that the statistical variability of system responses gains an obvious reduction,and the system performance improves significantly after the optimal polynomial controller is mounted.Using the control criterion with the minimum of the exceedance probability of system quantities in energy trade-off sense,the 1st-order controller suffices even for strongly nonlinear systems.This bypasses the need to utilize nonlinear controllers which may be associated with dynamical instabilities due to time delay and computational dynamics.Besides,the optimal polynomial controller exhibits high robustness to the levels of nonlinearity of Duffing oscillators.The classical statistical linearization based LQG,however,using the nominal Gaussian white noise as the external excitation does not have this advantage.
引文
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[5]Suhardjo J,Spencer Jr B F,Sain M K.Nonlinearoptimal control of a Duffing system[J].InternationalJournal of Non-linear Mechanics,1992,27(2):157—172.
[6]Yang J N,Agrawal A K,Chen S.Optimalpolynomial control for seismically excited non-linearand hysteretic structures[J].Earthquake Engineeringand Structural Dynamics,1996,25:1 211—1 230.
[7]Anderson Brian D O,Moore J.Optimal Control:Linear Quadratic Methods[M].Prentice-Hall,Englewood Cliffs,1990.
[8]Li J,Chen J B.The principle of preservation ofprobability and the generalized density evolutionequation[J].Structural Safety,2008,30:65—77.
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[10]Chen J B,Li J.Strategy for selecting representativepoints via tangent spheres in the probability densityevolution method[J].International Journal forNumerical Methods in Engineering,2008,74(13):1 988—2 014.
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