地震激励下20层钢结构基准模型的非线性随机最优控制
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摘要
建筑结构基准问题常用来比较各种振动控制策略的优劣。运用基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优(NSO)控制策略,研究了地震激励下20层钢结构基准模型的振动控制。建立基准模型的系统运动方程,通过模态变换转换到模态坐标下进行研究。由于结构的状态只有部分是可观测的,可通过分离原理将部分观测问题转化成完全可观测问题,由Kalman滤波方法得到系统状态的条件均值。运用拟可积Hamil-ton系统随机平均法得到随机平均方程,对部分模态进行控制求解。通过求解动态规划方程得到非线性的最优控制力,对结构的响应进行控制。将NSO控制得到的性能评价指标与线性二次型高斯(LQG)最优控制得到的评价指标进行对比,发现该非线性随机最优控制策略更加有效。
The benchmark problem of civil structures is always used to evaluate the efficacy of various structural control strategies.In this paper,a seismically excited 20-story steel structure benchmark model has been studied by using the nonlinear stochastic optimal(NSO) control strategy,which is based on both the stochastic averaging method for quasi Hamiltonian system and the stochastic dynamic programming principle.The dynamic equation of the benchmark problem has been established and transformed into modal coordinates.Because the state of the structure is only partially observed,the partially observation problem has been transformed into a completely observable problem by using the separation principle.The conditional mean of the system states have been estimated by the Kalman filter.The averaged equations have been obtained by using the stochastic averaging method for quasi-integrable Hamiltonian system.Part of modal has been taken to control.The nonlinear optimal control forces to control the response of the structure have been obtained by solving the dynamic programming equation.The performance indices of NSO control strategy have been compared with those of the linear quadratic Gaussian(LQG) control strategy.It is shown that the nonlinear stochastic optimal control strategy is more effective.
The benchmark problem of civil structures is always used to evaluate the efficacy of various structural control strategies.In this paper,a seismically excited 20-story steel structure benchmark model has been studied by using the nonlinear stochastic optimal(NSO) control strategy,which is based on both the stochastic averaging method for quasi Hamiltonian system and the stochastic dynamic programming principle.The dynamic equation of the benchmark problem has been established and transformed into modal coordinates.Because the state of the structure is only partially observed,the partially observation problem has been transformed into a completely observable problem by using the separation principle.The conditional mean of the system states have been estimated by the Kalman filter.The averaged equations have been obtained by using the stochastic averaging method for quasi-integrable Hamiltonian system.Part of modal has been taken to control.The nonlinear optimal control forces to control the response of the structure have been obtained by solving the dynamic programming equation.The performance indices of NSO control strategy have been compared with those of the linear quadratic Gaussian(LQG) control strategy.It is shown that the nonlinear stochastic optimal control strategy is more effective.
引文
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[2]Fleming W H,Rishel R W.Deterministic and Sto-chastic Optimal Control[M].Berlin:Springer-Ver-lag,1975.
[3]Fleming W H,Soner H M.Controlled MarkovProcesses and Viscosity Solutions[M].Berlin:Springer-Verlag,1992.
[4]Yong J M,Zhou X Y.Stochastic Controls,Hamilto-nian Systems and HJB Equations[M].New York:Springer-Verlag,1999.
[5]Bensoussan A.Stochastic Control of Partially Observ-able Systems[M].Cambridge:Cambridge UniversityPress,1992.
[6]Michael Athans.The role and use of the linear-quad-ratic-gaussian control system design[J].IEEE Trans-actions on Automatic Control,1971,16(6):529—552.
[7]Zhu W Q,Ying Z G.Optimal nonlinear feedback con-trol of quasi-Hamiltonian systems[J].Science in Chi-na Series A,1999,42(11):1 213—1 219.
[8]Zhu W Q,Ying Z G,Soong TT.Optimal nonlinearfeedback control of structures under random loading[A].Spencer BF,Johnson EA.Stochastic structuraldynamics[C].Rotterdam:Balkema,1998:141—148.
[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]Zhu W Q,Ying Z G,Ni Y Q,et al.Optimal nonlin-ear stochastic control of hysteretic systems[J].ASCEJournal of Engineering Mechanics,2000,126:1027—1 032.
[11]Zhu W Q,Ying Z G,Nonlinear stochastic optimalcontrol of partially observable linear structures[J].Engineering Structures,2002,24:333—342.
[12]Zhu W Q,Yang Y Q.Stochastic averaging of quasi-nonintegrable Hamiltonian systems[J].ASME Jour-nal of Applied Mechanics,1997,64:157—164.
[13]Zhu W Q,Huang Z L,Yang Y Q.Stochastic avera-ging of quasi-integrable Hamiltonian systems[J].ASME Journal of Applied Mechanics,1997,64(9):975—984.
[14]Spencer B F,Christenson R E,Dyke S J.Next gener-ation benchmark control problems for seismically ex-cited buildings[A].In:T.Kobori et al.Proceeding ofthe Second World conference on Structural Control[C].New York:Wiley,1999:1 135—1 360.
[15]Naeim F.Performance of 20extensively-instrumentedbuildings during the 1994Northridge earthquake[J].The Structural Design of Tall Buildings,1998,7(3):179—194.
[16]R W克拉夫,彭津,著.结构动力学[M].王光远,译.北京:科学出版社,1983.
[17]Edward J.Davison.A method for simplifying lineardynamic systems[J].IEEE Transactions On Automat-ic Control,1966,1:93—101.
[18]朱位秋.非线性随机动力学与控制:Hamilton理论体系框架[M].北京:科学出版社,2003.
[19]洪峰,江近仁,李玉亭.地震地面运动的功率谱模型及其参数的确定[J].地震工程与工程振动,1994,14(2):46—52.Hong Feng,Jiang Jinren,Li Yuting.Power spectralmodels of earthquake ground motions and evaluationof its parameters[J].Earthquake Engineering and En-gineering Vibration,1994,14(2):46—52.
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