基于正交展开方法的Duffing振子随机最优控制
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摘要
基于多维Hermite多项式的经典均相混沌展开,考察了Duffing振子随机最优多项式控制的正交展开方法,阐明了多项式系数演化与振子系统反应、最优控制力概率特性之间的联系。系统输入采用Karhunen-Loève展开表现的随机地震动。为降低混求解规模,引入位移-速度范数准则,发展了自适应混沌多项式展开策略。同时,基于Lyapunov稳定条件设计控制器的控制增益参数。数值算例分析表明,受控后系统位移和加速度的均方特征得到改善、振子系统的非线性程度减小,基于混沌多项式展开的最优控制方法能明显降低系统的随机涨落和显著改善系统的非线性反应性态。
An orthogonal expansion of stochastic optimal polynomial control,employing the homogenous chaos decomposition with multidimensional Hermite polynomials of random variable argument,of Duffing oscillators is investigated.It reveals the essential relationship between evolution of polynomial coefficients and probabilistic characteristics of oscillator response and control force.The procedure is demonstrated on a base-driven system whereby the ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen-Loève expansion.An adaptive scheme based on a displacement-velocity norm for stochastic approximation with polynomial chaos bases is proposed towards reducing computational effort,which is applied to the identification of phase orbits of nonlinear oscillators.This approximation is then integrated into the design of an optimal polynomial controller,allowing for the efficient estimation of statistics and probability density functions of quantities of interest.Numerical investigations are carried out employing the polynomial chaos expansion and the Lyapunov asymptotic stability condition based control policy.The results reveal that the performance,as gaged by probabilistic quantities of interest,of the controlled oscillators is greatly improved.It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.
An orthogonal expansion of stochastic optimal polynomial control,employing the homogenous chaos decomposition with multidimensional Hermite polynomials of random variable argument,of Duffing oscillators is investigated.It reveals the essential relationship between evolution of polynomial coefficients and probabilistic characteristics of oscillator response and control force.The procedure is demonstrated on a base-driven system whereby the ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen-Loève expansion.An adaptive scheme based on a displacement-velocity norm for stochastic approximation with polynomial chaos bases is proposed towards reducing computational effort,which is applied to the identification of phase orbits of nonlinear oscillators.This approximation is then integrated into the design of an optimal polynomial controller,allowing for the efficient estimation of statistics and probability density functions of quantities of interest.Numerical investigations are carried out employing the polynomial chaos expansion and the Lyapunov asymptotic stability condition based control policy.The results reveal that the performance,as gaged by probabilistic quantities of interest,of the controlled oscillators is greatly improved.It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.
引文
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[2]Ghanem R,Spanos P.Stochastic Finite Elements:ASpectral Approach[M].New York:Springer,1991.
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[4]Hover F S,Triantafyllou M S.Application ofpolynomial chaos in stability and control[J].Automatica,2006,42:789—795.
[5]Chen J B,Liu W Q,Peng Y B,et al.Stochasticseismic response and reliability analysis of base-isolated structures[J].Journal of EarthquakeEngineering,2007,11:903—924.
[6]彭勇波,李杰.非线性随机动力系统的最优多项式控制[J].振动工程学报,2010,23(4):366—372.
[7]Anderson Brian D O,Moore J.Optimal Control:Linear Quadratic Methods[M].Prentice-Hall,Englewood Cliffs,1990.
[8]Li J,Peng Y B,Chen J B.A physical approach tostructural stochastic optimal controls[J].Probabilistic Engineering Mechanics,2010,25:127—141.
[9]Loève M.Probability Theory[M].(4th Edition).New York:Springer,1978.
[10]Debusschere B J,Najm H N,Pébay P P,et al.Numerical challenges in the use of polynomial chaosrepresentations for stochastic processes[J].SIAMJournal on Scientific Computing,2004,26(2):698—719.
[11]Li R,Ghanem R.Adaptive polynomial chaosexpansions applied to statistics of extremes innonlinear random vibration[J].ProbabilisticEngineering Mechanics,1998,13(2):125—136.
[12]Yang J N,Li Z,Liu S C.Stable controllers forinstantaneous optimal control[J].ASCE Journal ofEngineering Mechanics,1992,118(8):1 612—1 630.
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