基于广义密度演化方程的结构随机最优控制
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摘要
基于广义密度演化方程和Pontryagin极大值原理,推导了一般随机激励作用下闭环系统随机最优控制中状态向量和控制力向量的物理解答,讨论了基于二阶统计量评价的控制律参数设计准则。以物理随机地震动模型为输入,考察了单层框架结构主动锚索系统的随机最优控制,并与经典LQG控制做了比较分析。结果表明,本文提出的随机最优控制方法具有适用性和有效性。
The celebrated Pontryagin's maximum principles is employed in this paper to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation.The optimal control scheme extends the classical stochastic optimal control methods,which is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes,and can govern the evolution details of stochastic dynamical systems,while the classical stochastic optimal control methods,such as the LQG control,essentially hold the system statistics,and cannot govern the desirable evolution details.Further,the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on the control criterion of system second-order statistics assessment.The stochastic optimal control of an active tendon control system,subjected to the random ground motion represented by the physical stochastic earthquake model is investigated.Numerical investigations reveal that the structural seismic performance is significantly improved when the optimal control strategy is applied.The LQG control,however,using the nominal Gaussian white noise as the external excitation cannot design the reasonable control system for civil engineering structures.It is indicated that the developed physical stochastic optimal control methodology has the validity and applicability.
The celebrated Pontryagin's maximum principles is employed in this paper to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation.The optimal control scheme extends the classical stochastic optimal control methods,which is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes,and can govern the evolution details of stochastic dynamical systems,while the classical stochastic optimal control methods,such as the LQG control,essentially hold the system statistics,and cannot govern the desirable evolution details.Further,the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on the control criterion of system second-order statistics assessment.The stochastic optimal control of an active tendon control system,subjected to the random ground motion represented by the physical stochastic earthquake model is investigated.Numerical investigations reveal that the structural seismic performance is significantly improved when the optimal control strategy is applied.The LQG control,however,using the nominal Gaussian white noise as the external excitation cannot design the reasonable control system for civil engineering structures.It is indicated that the developed physical stochastic optimal control methodology has the validity and applicability.
引文
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[14]Chung L L,Reinhorn A M,Soong T T.Experi-ments on active control of seismic structures[J].Journal of Engineering Mechanics,1988,114(2):241-256.
[2]Li J,Chen J B.Probability density evolution methodfor dynamic response analysis of structures with un-certain parameters[J].Computational Mechanics,2004,34(5):400-409.
[3]Li J,Chen J B.The principle of preservation of prob-ability and the generalized density evolution equation[J].Structural Safety,2008,30(1):65-77.
[4]Soong T T.Active Structural Control:Theory andPractice[M].Longman Scientific&Technical,NewYork,1990.
[5]Sperb R P.Maximum Principles and their Applica-tions[M].Academic Press,New York,1981.
[6]Athans M,Falb P.Optimal Control:An Introduc-tion to the Theory and its Applications[M].McGraw Hill,New York,1966.
[7]Zhang W S,Xu Y L.Closed form solution for along-wind response of actively controlled tall buildingswith LQG controllers[J].Journal of Wind Engi-neering and Industrial Aerodynamics,2001,89(9):785-807.
[8]Stengel R F,Ray LR and Marrison C I.Probabilisticevaluation of control system robustness[A].IMAWorkshop on Control Systems Design for AdvancedEngineering Systems:Complexity,Uncertainty,In-formation and Organization[C],Minneapolis,MN,1992.
[9]Zhu W Q,Ying Z G,Soong T T.An optimal nonlin-ear feedback control strategy for randomly excitedstructural systems[J].Nonlinear Dynamics,2001,24(1):31-51.
[10]Li J,Chen J B,Fan W L.The equivalent extreme-value event and evaluation of the structural systemreliability[J].Structural Safety,2007,29(2):112-131.
[11]李杰,艾晓秋.基于物理的随机地震动模型研究[J].地震工程与工程振动,2006,26(5):21-26.(LIJie,Ai Xiao-qiu.Study on random model of earth-quake ground motion based on physical process[J].Earthquake Engineering and Engineering Vibra-tion,2006,26(5):21-26.(in Chinese))
[12]陈建兵,李杰.随机结构反应概率密度演化分析的切球选点法[J].振动工程学报,2006,19(1):1-8.(CHEN Jian-bing,LI Jie.Strategy of selectingpoints via sphere of contact in probability density evo-lution method for response analysis of stochasticstructures[J].Journal of Vibration Engineering,2006,19(1):1-8.(in Chinese))
[13]Mathews J H,Fink K D,Fink K.Numerical Meth-ods using Matlab(4th Edition)[M].Prentice Hall,2003.
[14]Chung L L,Reinhorn A M,Soong T T.Experi-ments on active control of seismic structures[J].Journal of Engineering Mechanics,1988,114(2):241-256.
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