拟线性系统在宽带随机激励和多时滞反馈控制下的响应、稳定性及首次穿越时间
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摘要
本文将导师朱位秋院士提出的“随机激励的耗散的哈密顿系统理论”推广到具有多时滞反馈控制和宽带随机激励下的拟线性系统,研究了多时滞反馈控制和宽带随机激励下的拟线性系统的随机响应,概率为1稳定性,首次穿越时间以及建筑结构在地震作用下的时滞反馈控制的优化。随机响应研究中,将时滞的状态变量在平均意义上用无时滞的状态变量近似,由此得到了无时滞受控的拟可积哈密顿系统,再运用随机平均法,建立相应的Fokker-Planck-Kolmogorov(FPK)方程,求解该方程得到系统的响应,据此研究时滞反馈控制的控制效果;概率为1渐近稳定性研究中,先将时滞系统转化为非时滞系统,再用随机平均法得到关于系统慢变过程的It(?)随机微分方程,再引入新的范数,得到了最大Lyapunov指数的近似表达式,研究系统的概率为1渐近稳定性;首次穿越时间的研究中,先将时滞系统转化为非时滞系统,再用随机平均法得到系统慢变过程的It(?)随机微分方程,然后导出支配条件可靠性函数的后向Kolmotorov方程和支配平均首次穿越时间的Pontrygin方程及边值条件,求解这些方程得到系统的可靠性函数和平均首次穿越时间;在地震作用下建筑结构的时滞反馈控制优化研究中,先假定含待定增益的时滞速度反馈控制,利用随机平均法得到系统概率分布,然后以模态能量和控制力期望最小为性能指标,确定反馈控制增益。以上理论研究的结果和数值模拟结果完全吻合,研究表明时滞对系统的响应、稳定性及首次穿越时间都有不利的影响。但是,如果合理选取时滞时间,这些不利影响几乎可以完全消除。
In the present dissertation the theory of stochastically excited and dissipated Hamiltonian systems proposed by W.Q. Zhu is generalized to study the stochastic response, the stochastic stability and the reliability of quasi-linear system under multi-time-delayed feedback control and wide-band random excitations. In the study of the stochastic response, the system equations are transformed into differential equations without time delay and the averaged Ito stochastic differential equations for the slowly varying processes are derived. The stationary solution of the averaged FPK equation associated with the averaged It(o|^) equations is obtained and the effect of time-delayed feedback control on the responses is stuied. In the study of the asymptotic Lyapunov stability with probability 1, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. By introducing a new norm, the approximate formula for the largest Lyapunov exponent is derived. The necessary and sufficient condition for the asympototic Lyapunov stability with probability 1 is obtained. In the study of the first-passage failure, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. The conditional reliability function and moments of first-passage time are obtained from solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. In addition, a time-delayed feedback control problem of partially observable linear building structure under horizontal ground acceleration excitation is formulated and converted into that of completely observable linear structure by using the separation principle. The time-delayed control forces are approximately expressed in terms of control forces without time delay. The control system is then governed by Ito stochastic differential equations for the conditional means of system states and then transformed into those for the conditional means of modal energies by using the stochastic averaging method for quasi Hamiltonian systems. The control law is assumed to be modal velocity feedback control with time delay and the unknown control gains are determined by the modal performance indices. The comparising all the theoretic results and those from Monte-Carlo simulation shows that the two results in good agreement. Furthermore, it is shown that the time delay in feedback control affect the response, the asymptotic stability with probability 1 and the first-passage time remarkably. However, the deteriotation effect can be almost eliminated if the delay time is set correctly.
In the present dissertation the theory of stochastically excited and dissipated Hamiltonian systems proposed by W.Q. Zhu is generalized to study the stochastic response, the stochastic stability and the reliability of quasi-linear system under multi-time-delayed feedback control and wide-band random excitations. In the study of the stochastic response, the system equations are transformed into differential equations without time delay and the averaged Ito stochastic differential equations for the slowly varying processes are derived. The stationary solution of the averaged FPK equation associated with the averaged It(o|^) equations is obtained and the effect of time-delayed feedback control on the responses is stuied. In the study of the asymptotic Lyapunov stability with probability 1, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. By introducing a new norm, the approximate formula for the largest Lyapunov exponent is derived. The necessary and sufficient condition for the asympototic Lyapunov stability with probability 1 is obtained. In the study of the first-passage failure, the system equations are transformed into differential equations without time delay and the stochastic averaging method is used to derive the averaged Ito differential equations for the slow varing processes. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. The conditional reliability function and moments of first-passage time are obtained from solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. In addition, a time-delayed feedback control problem of partially observable linear building structure under horizontal ground acceleration excitation is formulated and converted into that of completely observable linear structure by using the separation principle. The time-delayed control forces are approximately expressed in terms of control forces without time delay. The control system is then governed by Ito stochastic differential equations for the conditional means of system states and then transformed into those for the conditional means of modal energies by using the stochastic averaging method for quasi Hamiltonian systems. The control law is assumed to be modal velocity feedback control with time delay and the unknown control gains are determined by the modal performance indices. The comparising all the theoretic results and those from Monte-Carlo simulation shows that the two results in good agreement. Furthermore, it is shown that the time delay in feedback control affect the response, the asymptotic stability with probability 1 and the first-passage time remarkably. However, the deteriotation effect can be almost eliminated if the delay time is set correctly.
引文
[1]朱位秋.非线性随机动力学与控制.北京:科学出版社,2004.
[2]朱位秋.随机振动.北京:科学出版社,1992.
[3]Einstein A.Investigation on the theory of Brownian Movement,in English Translation of Einstein Papers,Dover Publications,1956.
[4]Rice S O.Mathematical analysis of random noise,Bell Sys.Tech.J.23,1944,282-332;24,1945,46-156.Also in Selected Papers on Noise and Stochastic Processes,Wax,N.cal.,Dover,1954.
[5]Crandall S H.Random Vibration,The MIT Press,1958.
[6]Lin Y K.Probabilistic Theory of Structural Dynamics.New York:McGraw Hill,1967.
[7]Lin Y K,Cai G Q.Probabilistic Structural Dynamics:Advanced Theory and Applications.New York:McGraw Hill,1995.
[8]Nigam N C.Introduction to Random Vibrations.Cambridge:MIT Press,1983.
[9]Soong T T,Grigoriu M.Random Vibration of Mechanical and Structural Systems.New Jersey:Prentice Hall,1993.
[10]Caughey T K.Nonlinear Theory of Random Vibrations.In:Advances in Applied Mechanics,Yih C S(Ed.),New York:Academic Press,1971.
[11]Caughey T K.The behavior of linear systems with white noise input.Journal of Mathematical Physics,1962,32:2476-2479.
[12]Caughey T K.Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white noise excitation.Journal of the Acoustical Society of America,1967,35(11):1683-1692.
[13]Caughey T K,Ma F.The steady-state solution of a class of non-linear stochastic systems.International Journal of Non-Linear Mechanics,1982,17(3):137-142.
[14]Caughey T K,Ma F.The steady-state response of a class of dynamical systems to stochastic excitation,ASME Journal of Applied Mechanics,1982,49(3):629-632.
[15] Soize C. Steady-state solution of Fokker-Planck equation in higher dimension. Probabilistic Engineering Mechanics, 1988, 3(4): 196-206.
[16] Soize C. Exact stationary response of multi-dimensional non-linear Hamiltonian dynamical systems under parametric and external stochastic excitations. Journal of Sound and Vibration, 1991,149(1): 1-24.
[17] Soize C. The Fokker-Plank Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. Singapore: World Scientific, 1994.
[18] Lin Y K, Cai G Q. Exact stationary response solution for second order nonlinear systems under parametric and external white noise excitation. ASME Journal of Applied Mechanics, 1988, 55(2): 702-705.
[19] Yong Y, Lin Y K. Exact stationary response solution for second order nonlinear systems under parametric and external white-noise excitations. ASME Journal of Applied Mechanics, 1987, 54: 414-418.
[20] Dimentberg M F. An exact solution to a certain non-linear random vibration problem. International Journal of Non-Linear Mechanics, 1982,17(4): 231-236.
[21] Zhu W Q, Cai G Q, Lin Y K. On exact stationary solutions of stochastically perturbed Hamiltonian systems. Probabilistic Engineering Mechanics, 1990, 5:84-87.
[22] Zhu W Q, Cai G Q, Lin Y K, Stochastically perturbed Hamiltonian systems. Nonlinear Stochastic Mechanics, Proc. of IUTAM Symposium, Bellomo N, Casciati F. (Eds.), Spriger-Verlag, 1992. 543-552.
[23] Zhu W Q, Yang Y Q, Exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1996, 63(2): 493-500.
[24] Roberts J B, Spanos P D. Random Vibration and Statistical Linearization. New York: Wiley, 1990.
[25] Donley M G, Spanos P D. Dynamic Analysis of Non-Linear Structures by the Method of Statistical Quadratization. New York: Springer-Verlag, 1990.
[26] Caugher T K. On the response of nonlinear oscillators to stochastic excitation, Probabilistic Engineering. Mechanics, 1986, 1:2-4.
[27] Henning K, Roberts J B. Average methods for randomly excited nonlinear oscillators. Elishakoff I. and Lyon R. H. (Eds), Random Vibration-Status and recent Developments, Elsevier, 1986.
[28] Zhu W Q, Yu J S. The equivalent non-linear system method, Journal of Sound and Vibration, 1989,129: 385-395.
[29] Stratcnovitch R L. Topics in the Theory of Random Noise, Vol. 1. New York: Gordon Breach, 1963.
[30] Khasminskii R Z. A limit theorem for the solutions of differential equations with random right-band sides. Theory of Probability and Application, 1966, 11: 390-405.
[31] Namachchivaya N S, Lin Y K. Application of stochastic averaging for nonlinear dynamical systems with High damping. Probabilistic Engineering Mechanics. 1988, 3(3): 159-167.
[32] Zhu W Q, Yang Y Q, Stochastic averaging of quasi-nonintegrable-Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 157-164.
[33] Zhu W Q, Huang Z L, Yang Y Q. Stochastic averaging of quasi integrable-Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64:975-984.
[34] Zhu W Q, Huang Z L, Suzuke Y. Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. International Journal of Non-Linear Mechanics,2002, 37: 419-437.
[35] Crandall S H. Perturbation Techniques for random vibration of nonlinear systems. Journal of Acoustical Society of America, 1963, 35: 1700-1705.
[36] Zhu W Q, Luo M and Dong L. Semi-active control of Wind Excited Building Structures using MR/ER dampers, Probalibilistic Engineering Mechanics,2004, 19 (3), 279-285.
[37] Luo M and Zhu W Q, Nonlinear Stochastic Optimal Control of Offshore Platforms under Wave Loading, Journal of Sound and Vibration,2006, 296,734-745.
[38] Zhu W Q, Luo M and Ying Z G, Nonlinear stochastic optimal control of tall buildings under wind loading, Engineering Structures, 2004, 26(11), 1561-1572.
[39] Ni Y Q, Ying Z G, Wang J Y, Ko J M and Spencer B F, Stochastic Optimal Control of Wind-excitedTall Buildings using Semi-active MR-TLCDs, Probabilistic Engineering Mechanics, 2004, 79(3) , 269-277.
[40] Deng M L and Zhu W Q, Energy Diffusion Controlled Reaction Rate in Dissipative Hamiltonian Systems. Chinese Physics, 2007,16(06), 1009-1963.
[41] Deng M L and Zhu W Q, On the stochastic dynamics of molecular conformation, Journal of Zhejiang University Science A, 8(9): 1401-1407,2007.
[42] Deng M L and Zhu W Q,, Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise, European Physical Journal B 59(3): 391-397,2007.
[43] Chung L L, Lin C C, Lu K H. Time delay control of structures. Earthquake Engineering and Structural Dynamics, 1995, 24: 687-701
[44] Hu H Y, Wang Z H. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Berlin: Springer, 2002.
[45] Agrawal A K, Fujino Y, Bhartia B K. Instability duo to time delay and its compensation in active control of structures. Earthquake Engineering and Structural Dynamics, 1993, 22: 211-224.
[46] Soliman M A, Ray W H. Optimal feedback control for linear quadratic systems having time delays. International Journal of Control, 1972,15:609-615.
[47] Mutharasan R, Luus R. Analysis of time-delay systems by series approximations. American Institute of Chemical Engineers Journal, 1975,21: 567-572.
[48] Hammerstrom L G, Gros S K. Adaptation of optimal control theory to systems with time delays.International Journal of Control. 1980,32: 320-357.
[49] Chung L L, Lin R C, Soong T T, Reinborn A M. Experimental study of active control for MDOF seismic structures. ASCE Journal of Engineering Mechancis, 1989,115:1609-1627.
[50] Rodellar J, Chung L L, Soong T T, Reinhorn A M. Experimental digital control of structures. ASCE Journal of Engineering Mechanics, 1989,115:1245-1261.
[51] Inaudi J A, Kelly J M. A robust delay-compensation technique based on memory. Earthquake Engineering and Structural Dynamics, 1994,23:987-1001.
[52] Abdel M, Mooty, Roorda J. Time-delay compensation in active damping of structures. ASCE Journal of Engineering Mechanics, 1991,117: 2549-2570.
[53] Pu J P. Time delay compensation in active control of structure. Journal of Engineering Mechanics, 1998,124(9):1018-1028.
[54] Lin C C, Sheu J F, chu S Y. Time-delay effect and its solution for optimal output feedback control of structures. Earthquake Engineering and Structural Dynamics, 1996, 25:547-559.
[55] Maccari A. Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback. International Journal of Non-Linear Mechanics, 2003, 38:123-131.
[56] Hu H Y, Wang Z H. Stability analysis of damped SDOF systems with two time delays in state feedback. Journal of Sound and Vibration, 1998, 214(2): 213-225.
[57] Grigoriu M. Control of time delay linear system with Gaussian white noise. Probabilistic Engineering Mechanics, 1997,12(2): 89-96.
[58] Paola M Di, Pirrotta A. Time delay induced effects on control of linear systems under random excitation. Probabilistic Engineering Mechanics, 2001,16: 43-51.
[59] Fofana M S, Asymptotic stability of a stochastic delay equation. Probabilistic Engineering Mechanics, 2002,17:385-392.
[60] Liu Z H, Zhu W Q, Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control, Journal of Sound and Vibration ,2007, 299,178-195..
[61] Kozin F. A survey of stability of stochastic systems. Automatica, 1969,5:95-112
[62] Khasminskii R Z, Stochastic stability of differential equations, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980.
[63] Oseledec V I. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Transactions of Moscow Mathematics Society, 1968, 19:197-231.
[64] Zhu W Q, Huang Z L. Stochastic stability of quasi-non-integrable Hamiltonian systems. Journal of Sound and Vibration, 1998, 218(5):769-789.
[65] Zhu W Q, Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems.International Journal of Non-Linear Mechanics,2004,39:569-579.
[66]Zhu W Q,Huang Z L.Lyapunov exponents and stochastic stability of quasi-integrable Hamiltonian systems.ASME,Journal of Applied Mechanics,1999,66:211-217.
[67]Huang Z L,Zhu W Q,Ni Y Q,Ko J M.Stochastic averaging of strongly nonlinear oscillators under bounded noise excitation.Journal of Sound and Vibration,2002,54(2):245-267.
[68]Zhu W Q,Huang Z L,Suzuki Y.Stochastic averaging and Lyapunov exponent of quasi partially quasi-integrable Hamiltonian systems.International Journal of Non-Linear Dynamics,2002,37:419-437.
[69]Zhu W Q,Huang Z L,Suzuki Y.Response and stability of strongly non-linear oscillators under wide-band random excitations.International Journal of Non-Linear Mechanics,2001,36:1235-1250.
[70]Huang Z L,Zhu W Q.Lyapunov exponents and almost sure asymptotic stability of quasi-linear gyroscopic systems.International Journal of Non-Linear mechanics,2000,35:645-655.
[71]Huang Z L,Zhu W Q,A New approach to almost-sure asymptotic stability of stochastic systems of higher dimension,International Journal of Non-Linear mechanics,2003,38:239-247.
[72]Franklin J Nand Rodemich,E.R.,Numerical analysis of an elliptic parabolic partial differentia equation,SIAM J.Num.Anal.,5(1968),680-716.
[73]Kozin,F.,First passage times-some results,Proc.Int.Workshop on Stochastic Struct.Mech.,Rep.1-83,Universitat Innsbruck,1983.
[74]Dynkin,E.B.,Markov Processes,Vols.1 and 2.,Springer-Verlag,1965.
[75]#12
[76]Khasminskii R.Z.Averaging principle for stochastic differential Ito equations.Kibernetia,1968,4:260-279(in Russian).
[77]Zhu W Q and Lei Y,First passage time for state transition of randomly excited systems,Bulletin of The International Statistical Institute,Proceedings of the 47~(th)Session,Vol.LⅢ,book 3(Invited Papers),Aug.27-Sept.6,1989,517-531.
[78]Roberts J B,1976,First Passage Probability for Nonlinear Oscillator,ASCE Journal of Engineering Mechanics Division,102,pp.851-866.
[79]Cai G.Q and Lin Y K,1994,"On Statistics of First-Passage Failure",ASME Journal of Applied Mechanics,61,pp.93-99.
[80]朱位秋,黄志龙,随机激励的耗散的Hamilton系统理论的研究进展,力学进展,2000,Vol.30,No.4,481-494.
[81]Zhu W Q,Huang,Z L and Suzuki,Y,Response and stability of strongly non-linear oscillators under wide-band random excitation,International Journal of Non-Linear Mechanics 36(2001)1235-1250.
[82]Huang Z L,Zhu W Q,Exact stationary solutions of averaged equation of stochastically and harmonicially excited MDOF quasi-linear systems with internal and/or external resonances.Journal of Sound and Vibration,1997,204:249-258.
[83]Huang Z L,Zhu W Q,Suzuki Y,Stochstic averaging of strongly non-linear oscillators under combined harmonic and white noise excitation.Journal of Sound ans Vibration,200,238:233-256.
[84]Huang Z L,Zhu W Q,Ni Y Q,Ko J M,Stochstic averaging of strongly non-linear oscillators under bounded noise excitation.Journal of Sound and Vibration,2002,254:245-267.
[85]Gan C B and Zhu W Q,First-passage failure of quasi non-integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,36(2),2001,209-220.
[86]Zhu W Q,Huang Z L and Deng M L,Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,37(6),2002,1057-1071.
[87]Zhu W Q,Huang Z L and Deng M L,First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,38(8),2003,1133-1148.
[88]Zhu W Q and Wu Y J,Optimal bounded control of first-passage failure of strongly non-linear oscillators under combined harmonic and white-noise excitations,Journal of Sound and Vibration,271(1-2),2004,Pages 83-101.
[89]Agrawal A K and Yang J N,Effect of fixed time delay on stability and performance of actively controlled civil engineering structures,Earthquake Engineering and Structural Dynamics 26,1997,1169-1185.
[90]Balanow A G,Janson N B,Scholl,E.Control of noise-induced oscillations by delayed feedback,Phycica D 199,2004,1-12.
[91]Di Paola M and Pirrotta A,Time delay induced effects on control of linear systems under random excitation,Probabilistic Engineering Mechanics 16,2001,43-51.
[92]Hu H Y and Wang Z H,Dynamics of Controlled Mechanical Systems with Delayed Feedback,Springer-Verlag,Berlin,2002.
[93]Kuo B C,Automatic Control Systems,Prentice-Hall,Englewood Cliffs,NJ,1987.
[94]Liu Z H and Zhu W Q,Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control,Journal of Sound and Vibration 299,2007,178-195.
[95]Malek-Zavarei M and Jamshidi M,Time-Delay Systems:Analysis,Optimization and Applications,North-Holland,Now York,1987.
[96]Robers J B and Spanos P D,Stochastic averaging:an approximate method of solving random vibration problems.International Journal of Non-linear Mechanics,21,1986,111-134.
[97]Stratonovich R L,Topics in the Theory of Random Noise,Vol.2.New York:Gordon and breach,1967.
[98]Zhu W Q,Stochastic averaging methods in random vibration,ASME Applied Mechanics Reviews,41,1988,189-199.
[99]Kozin F,A survey of stability of stochastic systems.Automatica,1969,5:95-112.
[100]Khasminskii R Z,Stochastic stability of differential equations,,Alphen aan den Rijn:Sijthoff& Noordhoff,1980.
[101]Oseledec V I,A multiplicative ergodic theorem,Lyapunov characteristic numbers for dynamical systems.Transactions of Moscow Mathematics Society,1968,19:197-231.
[102]Khasminskii R Z.Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems.Theory of Probability and Application,1967,11:144-147.
[103]Arnold L.A Formula connecting sample and moment stability of linear stochastic systems.SIAM Journal of Applied Mathematics,1984,44:793-802.
[104]Doyle M M,Sri Namachchivaya N,Arnold L,Small noise expansion of moment Lyapunov exponents for two-dimensional systems.Advances in Nonlinear Stochastic Mechnics,Naess A,Krenk,S(Eds.),Dordrecht:Kluwer Academic Publishers,1996,153-168.
[105]Sri Mamachchivaya N,Van Rossel H J,Doyle M M,Moment Lyapunov exponent for two coupled oscillators driven by real noise,SIAM Journal of Applied Mathematics,1996,56:1400-1423.
[106]Liu Z H and Zhu W Q.Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control.Journal of Sound and Vibration 2007;299:178-195.
[107]Liu Z H and Zhu W Q,Stability and response of quasi-integrable Hamiltonian systems with time-delayed feedback control.IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty,H.Y.Hu & E.Kreuzer (Eds.),Springer,2007:383-388.
[108]邓茂林,具有控制的拟Hamilton系统的首次穿越时间.浙江大学硕士学位论文.2002.
[109]Housner G W,Bergman LA,Caughey T K,Chassiakos A G,Claus R O,Masri S F,Skelton RE,Soong T T,Spence B F and Yao J T P.Structural control:past,present,and future.Joural of Engineering Mechanics ASCE,1997,123(9):897-971.
[110]Yao J T P.Concept of structural control.Journal of Structural Division ASCE 1972,98:1567-1574.
[111]Fleming W H,Soner,H.M.,Controlled Markov processes and viscosity solutions,New York:Springer,1993.
[112]Yong J M and Zhou X Y,Stochastic control Hamiltonian systems and HJB equation.New York:Springer,1999.
[113]Zhu W Q,Huang Z L Yang Y Q,Stochastic averaging of quasi intcgrable Hamiltonian systems.ASME Journal of Applied Mechanics,1997,64:975-984.
[114]Yong J M and Zhou X Y,Stochastic control Hamiltonian systems and HJB equation.New York:Springer,1999.
[115]Zhu W Q and Ying Z G,Optimal nonlinear feedback control of quasi Hamiltonian systems.Science in China,Series A,1999,42:1213-1219.
[116]Zhu W Q,Ying Z G.and Soong T T,An optimal nonlinear feedback control strategy for randomly excited structural systems.Nonlinear Dynamics,2001,24:31-51.
[117]Soong T T.Active structural control:theory and practice.Longman Wiley,London,England,1990.
[118]Choksy N H,Time lag systems.Progress in Control Engineering,19621:17-38
[119]Sain P M,Spencer B F,Sain M K,Suhardjo J,Structural control design in the presence of time delays.Proceedings of the 9th Engineering Mechanics Conference,ASCE,College Station,1992,24-27.
[120]Agrawal A K,Yang J N,Effect of fixed time delay on stability and performance of actively controlled civil engineering structures.Earthquake Engineering and Structural,1997,26:1169-1185.
[121]Cai G,Huang J,Optimal control method for seismically excited building structures with time delay in control.Journal of Engineering Mechanics,2002,128(6):602-12.
[122]Firdaus E,Hubertus V B,Phailaung P,Time-delayed control design for active control of structures:principles and applications.Structural control and health monitoring,2007,14:27-61.
[123]欧进萍,王光远.结构随机振动,北京,高等教育出版社,1995.
[124]Fleming W H,Rishel R W,Deterministic and stochastic optimal control,Berlin, Spring-Verlag, (1975).
[125]Bensoussan A, Stochastic control of partially observable systems, Cambridge: Cambridge University Press, (1992).
[126]Li X P, Ying Z G, Feedback control optimization problems for seismically excited buildings. Acta Mechanica Solida Sinica, 20, 4, 2007.
[2]朱位秋.随机振动.北京:科学出版社,1992.
[3]Einstein A.Investigation on the theory of Brownian Movement,in English Translation of Einstein Papers,Dover Publications,1956.
[4]Rice S O.Mathematical analysis of random noise,Bell Sys.Tech.J.23,1944,282-332;24,1945,46-156.Also in Selected Papers on Noise and Stochastic Processes,Wax,N.cal.,Dover,1954.
[5]Crandall S H.Random Vibration,The MIT Press,1958.
[6]Lin Y K.Probabilistic Theory of Structural Dynamics.New York:McGraw Hill,1967.
[7]Lin Y K,Cai G Q.Probabilistic Structural Dynamics:Advanced Theory and Applications.New York:McGraw Hill,1995.
[8]Nigam N C.Introduction to Random Vibrations.Cambridge:MIT Press,1983.
[9]Soong T T,Grigoriu M.Random Vibration of Mechanical and Structural Systems.New Jersey:Prentice Hall,1993.
[10]Caughey T K.Nonlinear Theory of Random Vibrations.In:Advances in Applied Mechanics,Yih C S(Ed.),New York:Academic Press,1971.
[11]Caughey T K.The behavior of linear systems with white noise input.Journal of Mathematical Physics,1962,32:2476-2479.
[12]Caughey T K.Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white noise excitation.Journal of the Acoustical Society of America,1967,35(11):1683-1692.
[13]Caughey T K,Ma F.The steady-state solution of a class of non-linear stochastic systems.International Journal of Non-Linear Mechanics,1982,17(3):137-142.
[14]Caughey T K,Ma F.The steady-state response of a class of dynamical systems to stochastic excitation,ASME Journal of Applied Mechanics,1982,49(3):629-632.
[15] Soize C. Steady-state solution of Fokker-Planck equation in higher dimension. Probabilistic Engineering Mechanics, 1988, 3(4): 196-206.
[16] Soize C. Exact stationary response of multi-dimensional non-linear Hamiltonian dynamical systems under parametric and external stochastic excitations. Journal of Sound and Vibration, 1991,149(1): 1-24.
[17] Soize C. The Fokker-Plank Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. Singapore: World Scientific, 1994.
[18] Lin Y K, Cai G Q. Exact stationary response solution for second order nonlinear systems under parametric and external white noise excitation. ASME Journal of Applied Mechanics, 1988, 55(2): 702-705.
[19] Yong Y, Lin Y K. Exact stationary response solution for second order nonlinear systems under parametric and external white-noise excitations. ASME Journal of Applied Mechanics, 1987, 54: 414-418.
[20] Dimentberg M F. An exact solution to a certain non-linear random vibration problem. International Journal of Non-Linear Mechanics, 1982,17(4): 231-236.
[21] Zhu W Q, Cai G Q, Lin Y K. On exact stationary solutions of stochastically perturbed Hamiltonian systems. Probabilistic Engineering Mechanics, 1990, 5:84-87.
[22] Zhu W Q, Cai G Q, Lin Y K, Stochastically perturbed Hamiltonian systems. Nonlinear Stochastic Mechanics, Proc. of IUTAM Symposium, Bellomo N, Casciati F. (Eds.), Spriger-Verlag, 1992. 543-552.
[23] Zhu W Q, Yang Y Q, Exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1996, 63(2): 493-500.
[24] Roberts J B, Spanos P D. Random Vibration and Statistical Linearization. New York: Wiley, 1990.
[25] Donley M G, Spanos P D. Dynamic Analysis of Non-Linear Structures by the Method of Statistical Quadratization. New York: Springer-Verlag, 1990.
[26] Caugher T K. On the response of nonlinear oscillators to stochastic excitation, Probabilistic Engineering. Mechanics, 1986, 1:2-4.
[27] Henning K, Roberts J B. Average methods for randomly excited nonlinear oscillators. Elishakoff I. and Lyon R. H. (Eds), Random Vibration-Status and recent Developments, Elsevier, 1986.
[28] Zhu W Q, Yu J S. The equivalent non-linear system method, Journal of Sound and Vibration, 1989,129: 385-395.
[29] Stratcnovitch R L. Topics in the Theory of Random Noise, Vol. 1. New York: Gordon Breach, 1963.
[30] Khasminskii R Z. A limit theorem for the solutions of differential equations with random right-band sides. Theory of Probability and Application, 1966, 11: 390-405.
[31] Namachchivaya N S, Lin Y K. Application of stochastic averaging for nonlinear dynamical systems with High damping. Probabilistic Engineering Mechanics. 1988, 3(3): 159-167.
[32] Zhu W Q, Yang Y Q, Stochastic averaging of quasi-nonintegrable-Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 157-164.
[33] Zhu W Q, Huang Z L, Yang Y Q. Stochastic averaging of quasi integrable-Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64:975-984.
[34] Zhu W Q, Huang Z L, Suzuke Y. Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. International Journal of Non-Linear Mechanics,2002, 37: 419-437.
[35] Crandall S H. Perturbation Techniques for random vibration of nonlinear systems. Journal of Acoustical Society of America, 1963, 35: 1700-1705.
[36] Zhu W Q, Luo M and Dong L. Semi-active control of Wind Excited Building Structures using MR/ER dampers, Probalibilistic Engineering Mechanics,2004, 19 (3), 279-285.
[37] Luo M and Zhu W Q, Nonlinear Stochastic Optimal Control of Offshore Platforms under Wave Loading, Journal of Sound and Vibration,2006, 296,734-745.
[38] Zhu W Q, Luo M and Ying Z G, Nonlinear stochastic optimal control of tall buildings under wind loading, Engineering Structures, 2004, 26(11), 1561-1572.
[39] Ni Y Q, Ying Z G, Wang J Y, Ko J M and Spencer B F, Stochastic Optimal Control of Wind-excitedTall Buildings using Semi-active MR-TLCDs, Probabilistic Engineering Mechanics, 2004, 79(3) , 269-277.
[40] Deng M L and Zhu W Q, Energy Diffusion Controlled Reaction Rate in Dissipative Hamiltonian Systems. Chinese Physics, 2007,16(06), 1009-1963.
[41] Deng M L and Zhu W Q, On the stochastic dynamics of molecular conformation, Journal of Zhejiang University Science A, 8(9): 1401-1407,2007.
[42] Deng M L and Zhu W Q,, Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise, European Physical Journal B 59(3): 391-397,2007.
[43] Chung L L, Lin C C, Lu K H. Time delay control of structures. Earthquake Engineering and Structural Dynamics, 1995, 24: 687-701
[44] Hu H Y, Wang Z H. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Berlin: Springer, 2002.
[45] Agrawal A K, Fujino Y, Bhartia B K. Instability duo to time delay and its compensation in active control of structures. Earthquake Engineering and Structural Dynamics, 1993, 22: 211-224.
[46] Soliman M A, Ray W H. Optimal feedback control for linear quadratic systems having time delays. International Journal of Control, 1972,15:609-615.
[47] Mutharasan R, Luus R. Analysis of time-delay systems by series approximations. American Institute of Chemical Engineers Journal, 1975,21: 567-572.
[48] Hammerstrom L G, Gros S K. Adaptation of optimal control theory to systems with time delays.International Journal of Control. 1980,32: 320-357.
[49] Chung L L, Lin R C, Soong T T, Reinborn A M. Experimental study of active control for MDOF seismic structures. ASCE Journal of Engineering Mechancis, 1989,115:1609-1627.
[50] Rodellar J, Chung L L, Soong T T, Reinhorn A M. Experimental digital control of structures. ASCE Journal of Engineering Mechanics, 1989,115:1245-1261.
[51] Inaudi J A, Kelly J M. A robust delay-compensation technique based on memory. Earthquake Engineering and Structural Dynamics, 1994,23:987-1001.
[52] Abdel M, Mooty, Roorda J. Time-delay compensation in active damping of structures. ASCE Journal of Engineering Mechanics, 1991,117: 2549-2570.
[53] Pu J P. Time delay compensation in active control of structure. Journal of Engineering Mechanics, 1998,124(9):1018-1028.
[54] Lin C C, Sheu J F, chu S Y. Time-delay effect and its solution for optimal output feedback control of structures. Earthquake Engineering and Structural Dynamics, 1996, 25:547-559.
[55] Maccari A. Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback. International Journal of Non-Linear Mechanics, 2003, 38:123-131.
[56] Hu H Y, Wang Z H. Stability analysis of damped SDOF systems with two time delays in state feedback. Journal of Sound and Vibration, 1998, 214(2): 213-225.
[57] Grigoriu M. Control of time delay linear system with Gaussian white noise. Probabilistic Engineering Mechanics, 1997,12(2): 89-96.
[58] Paola M Di, Pirrotta A. Time delay induced effects on control of linear systems under random excitation. Probabilistic Engineering Mechanics, 2001,16: 43-51.
[59] Fofana M S, Asymptotic stability of a stochastic delay equation. Probabilistic Engineering Mechanics, 2002,17:385-392.
[60] Liu Z H, Zhu W Q, Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control, Journal of Sound and Vibration ,2007, 299,178-195..
[61] Kozin F. A survey of stability of stochastic systems. Automatica, 1969,5:95-112
[62] Khasminskii R Z, Stochastic stability of differential equations, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980.
[63] Oseledec V I. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Transactions of Moscow Mathematics Society, 1968, 19:197-231.
[64] Zhu W Q, Huang Z L. Stochastic stability of quasi-non-integrable Hamiltonian systems. Journal of Sound and Vibration, 1998, 218(5):769-789.
[65] Zhu W Q, Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems.International Journal of Non-Linear Mechanics,2004,39:569-579.
[66]Zhu W Q,Huang Z L.Lyapunov exponents and stochastic stability of quasi-integrable Hamiltonian systems.ASME,Journal of Applied Mechanics,1999,66:211-217.
[67]Huang Z L,Zhu W Q,Ni Y Q,Ko J M.Stochastic averaging of strongly nonlinear oscillators under bounded noise excitation.Journal of Sound and Vibration,2002,54(2):245-267.
[68]Zhu W Q,Huang Z L,Suzuki Y.Stochastic averaging and Lyapunov exponent of quasi partially quasi-integrable Hamiltonian systems.International Journal of Non-Linear Dynamics,2002,37:419-437.
[69]Zhu W Q,Huang Z L,Suzuki Y.Response and stability of strongly non-linear oscillators under wide-band random excitations.International Journal of Non-Linear Mechanics,2001,36:1235-1250.
[70]Huang Z L,Zhu W Q.Lyapunov exponents and almost sure asymptotic stability of quasi-linear gyroscopic systems.International Journal of Non-Linear mechanics,2000,35:645-655.
[71]Huang Z L,Zhu W Q,A New approach to almost-sure asymptotic stability of stochastic systems of higher dimension,International Journal of Non-Linear mechanics,2003,38:239-247.
[72]Franklin J Nand Rodemich,E.R.,Numerical analysis of an elliptic parabolic partial differentia equation,SIAM J.Num.Anal.,5(1968),680-716.
[73]Kozin,F.,First passage times-some results,Proc.Int.Workshop on Stochastic Struct.Mech.,Rep.1-83,Universitat Innsbruck,1983.
[74]Dynkin,E.B.,Markov Processes,Vols.1 and 2.,Springer-Verlag,1965.
[75]#12
[76]Khasminskii R.Z.Averaging principle for stochastic differential Ito equations.Kibernetia,1968,4:260-279(in Russian).
[77]Zhu W Q and Lei Y,First passage time for state transition of randomly excited systems,Bulletin of The International Statistical Institute,Proceedings of the 47~(th)Session,Vol.LⅢ,book 3(Invited Papers),Aug.27-Sept.6,1989,517-531.
[78]Roberts J B,1976,First Passage Probability for Nonlinear Oscillator,ASCE Journal of Engineering Mechanics Division,102,pp.851-866.
[79]Cai G.Q and Lin Y K,1994,"On Statistics of First-Passage Failure",ASME Journal of Applied Mechanics,61,pp.93-99.
[80]朱位秋,黄志龙,随机激励的耗散的Hamilton系统理论的研究进展,力学进展,2000,Vol.30,No.4,481-494.
[81]Zhu W Q,Huang,Z L and Suzuki,Y,Response and stability of strongly non-linear oscillators under wide-band random excitation,International Journal of Non-Linear Mechanics 36(2001)1235-1250.
[82]Huang Z L,Zhu W Q,Exact stationary solutions of averaged equation of stochastically and harmonicially excited MDOF quasi-linear systems with internal and/or external resonances.Journal of Sound and Vibration,1997,204:249-258.
[83]Huang Z L,Zhu W Q,Suzuki Y,Stochstic averaging of strongly non-linear oscillators under combined harmonic and white noise excitation.Journal of Sound ans Vibration,200,238:233-256.
[84]Huang Z L,Zhu W Q,Ni Y Q,Ko J M,Stochstic averaging of strongly non-linear oscillators under bounded noise excitation.Journal of Sound and Vibration,2002,254:245-267.
[85]Gan C B and Zhu W Q,First-passage failure of quasi non-integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,36(2),2001,209-220.
[86]Zhu W Q,Huang Z L and Deng M L,Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,37(6),2002,1057-1071.
[87]Zhu W Q,Huang Z L and Deng M L,First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems,International Journal of Non-Linear Mechanics,38(8),2003,1133-1148.
[88]Zhu W Q and Wu Y J,Optimal bounded control of first-passage failure of strongly non-linear oscillators under combined harmonic and white-noise excitations,Journal of Sound and Vibration,271(1-2),2004,Pages 83-101.
[89]Agrawal A K and Yang J N,Effect of fixed time delay on stability and performance of actively controlled civil engineering structures,Earthquake Engineering and Structural Dynamics 26,1997,1169-1185.
[90]Balanow A G,Janson N B,Scholl,E.Control of noise-induced oscillations by delayed feedback,Phycica D 199,2004,1-12.
[91]Di Paola M and Pirrotta A,Time delay induced effects on control of linear systems under random excitation,Probabilistic Engineering Mechanics 16,2001,43-51.
[92]Hu H Y and Wang Z H,Dynamics of Controlled Mechanical Systems with Delayed Feedback,Springer-Verlag,Berlin,2002.
[93]Kuo B C,Automatic Control Systems,Prentice-Hall,Englewood Cliffs,NJ,1987.
[94]Liu Z H and Zhu W Q,Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control,Journal of Sound and Vibration 299,2007,178-195.
[95]Malek-Zavarei M and Jamshidi M,Time-Delay Systems:Analysis,Optimization and Applications,North-Holland,Now York,1987.
[96]Robers J B and Spanos P D,Stochastic averaging:an approximate method of solving random vibration problems.International Journal of Non-linear Mechanics,21,1986,111-134.
[97]Stratonovich R L,Topics in the Theory of Random Noise,Vol.2.New York:Gordon and breach,1967.
[98]Zhu W Q,Stochastic averaging methods in random vibration,ASME Applied Mechanics Reviews,41,1988,189-199.
[99]Kozin F,A survey of stability of stochastic systems.Automatica,1969,5:95-112.
[100]Khasminskii R Z,Stochastic stability of differential equations,,Alphen aan den Rijn:Sijthoff& Noordhoff,1980.
[101]Oseledec V I,A multiplicative ergodic theorem,Lyapunov characteristic numbers for dynamical systems.Transactions of Moscow Mathematics Society,1968,19:197-231.
[102]Khasminskii R Z.Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems.Theory of Probability and Application,1967,11:144-147.
[103]Arnold L.A Formula connecting sample and moment stability of linear stochastic systems.SIAM Journal of Applied Mathematics,1984,44:793-802.
[104]Doyle M M,Sri Namachchivaya N,Arnold L,Small noise expansion of moment Lyapunov exponents for two-dimensional systems.Advances in Nonlinear Stochastic Mechnics,Naess A,Krenk,S(Eds.),Dordrecht:Kluwer Academic Publishers,1996,153-168.
[105]Sri Mamachchivaya N,Van Rossel H J,Doyle M M,Moment Lyapunov exponent for two coupled oscillators driven by real noise,SIAM Journal of Applied Mathematics,1996,56:1400-1423.
[106]Liu Z H and Zhu W Q.Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control.Journal of Sound and Vibration 2007;299:178-195.
[107]Liu Z H and Zhu W Q,Stability and response of quasi-integrable Hamiltonian systems with time-delayed feedback control.IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty,H.Y.Hu & E.Kreuzer (Eds.),Springer,2007:383-388.
[108]邓茂林,具有控制的拟Hamilton系统的首次穿越时间.浙江大学硕士学位论文.2002.
[109]Housner G W,Bergman LA,Caughey T K,Chassiakos A G,Claus R O,Masri S F,Skelton RE,Soong T T,Spence B F and Yao J T P.Structural control:past,present,and future.Joural of Engineering Mechanics ASCE,1997,123(9):897-971.
[110]Yao J T P.Concept of structural control.Journal of Structural Division ASCE 1972,98:1567-1574.
[111]Fleming W H,Soner,H.M.,Controlled Markov processes and viscosity solutions,New York:Springer,1993.
[112]Yong J M and Zhou X Y,Stochastic control Hamiltonian systems and HJB equation.New York:Springer,1999.
[113]Zhu W Q,Huang Z L Yang Y Q,Stochastic averaging of quasi intcgrable Hamiltonian systems.ASME Journal of Applied Mechanics,1997,64:975-984.
[114]Yong J M and Zhou X Y,Stochastic control Hamiltonian systems and HJB equation.New York:Springer,1999.
[115]Zhu W Q and Ying Z G,Optimal nonlinear feedback control of quasi Hamiltonian systems.Science in China,Series A,1999,42:1213-1219.
[116]Zhu W Q,Ying Z G.and Soong T T,An optimal nonlinear feedback control strategy for randomly excited structural systems.Nonlinear Dynamics,2001,24:31-51.
[117]Soong T T.Active structural control:theory and practice.Longman Wiley,London,England,1990.
[118]Choksy N H,Time lag systems.Progress in Control Engineering,19621:17-38
[119]Sain P M,Spencer B F,Sain M K,Suhardjo J,Structural control design in the presence of time delays.Proceedings of the 9th Engineering Mechanics Conference,ASCE,College Station,1992,24-27.
[120]Agrawal A K,Yang J N,Effect of fixed time delay on stability and performance of actively controlled civil engineering structures.Earthquake Engineering and Structural,1997,26:1169-1185.
[121]Cai G,Huang J,Optimal control method for seismically excited building structures with time delay in control.Journal of Engineering Mechanics,2002,128(6):602-12.
[122]Firdaus E,Hubertus V B,Phailaung P,Time-delayed control design for active control of structures:principles and applications.Structural control and health monitoring,2007,14:27-61.
[123]欧进萍,王光远.结构随机振动,北京,高等教育出版社,1995.
[124]Fleming W H,Rishel R W,Deterministic and stochastic optimal control,Berlin, Spring-Verlag, (1975).
[125]Bensoussan A, Stochastic control of partially observable systems, Cambridge: Cambridge University Press, (1992).
[126]Li X P, Ying Z G, Feedback control optimization problems for seismically excited buildings. Acta Mechanica Solida Sinica, 20, 4, 2007.