不确定拟Hamilton系统的非线性随机最优控制的鲁棒性与其鲁棒控制
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摘要
本文由两部分组成,分别研究不确定拟Hamilton系统的非线性随机最优控制的鲁棒性与其鲁棒控制。在第一部分中,首先基于不确定参数和随机激励的独立性,应用随机平均法和随机动态规划原理得到具平均值参数名义拟Hamilton系统的非线性随机最优控制;然后,应用随机平均法和概率分析得到不确定拟Hamilton系统均方根响应、控制效果和控制效率的均值和标准差;最后,引进受控均方根响应、控制效果和控制效率的变差系数对不确定参数变差系数的敏感性作为鲁棒性评价指标,对非线性随机最优控制的鲁棒性进行分析。作为拟Hamilton系统非线性随机最优控制策略在实际工程中的应用,给出了Bouc-Wen及Preisach滞迟系统两个例子,并使用上述的鲁棒性分析方法对它们进行全面研究。数值结果显示了这一控制方法的卓越的鲁棒性能。
在第二部分中,基于随机平均法和随机微分对策,提出了受有界的参数和外部扰动的拟Hamilton系统的极小极大最优控制策略。寻求最坏情况下最优控制的目标通过求解一个随机微分对策问题实现,而最坏情况扰动与相应的最优控制由Hamilton-Jacobi-Isaacs方程确定。作为上述方法的一个推广,详细阐述了其在反馈稳定化问题中的应用。通过求解一个具待定成本函数的随机微分对策问题确定系统的最坏情况扰动及相应的最优控制的形式,然后以最坏扰动下最优控制系统的最大Lyapunov指数最小为准则确定成本函数。
The present dissertation consists of two parts. In the first part, the robustness of the nonlinear stochastic optimal control for uncertain quasi Hamiltonian systems is studied. Based on the independence of uncertain parameters and stochastic excitations, the nonlinear stochastic optimal control for the nominal quasi Hamiltonian system with average-value parameters is firstly obtained by using the stochastic averaging method and stochastic dynamical programming principle. Then, the means and standard deviations of root-mean-square responses, control effectiveness and control efficiency for the uncertain quasi Hamiltonian system are calculated by using the stochastic averaging method and the probabilistic analysis. By introducing the sensitivities of the variation coefficients of controlled root-mean-square responses, control effectiveness and control efficiency to those of the uncertain parameters as measures, the robustness of the nonlinear stochastic optimal control is evaluated. As the application of the nonlinear stochastic optimal control strategy, controlled Bouc-Wen and Preisach hysteretic systems are studied and the robustness of the control strategy is also analyzed. Numerical results show the remarkable robustness of nonlinear stochastic optimal control.
In the second part, the robust control of uncertain quasi Hamiltonian systems is studied. A minimax optimal control strategy for quasi Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. The proposed strategy is searching for the optimal worst-case controller by solving a stochastic differential game problem, and the worst-case disturbances and the corresponding optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs equation. The feedback stabilization of uncertain quasi Hamiltonian systems are investigated as a generalization of the above robust control strategy. The form of the worst-case disturbances and the optimal controls are firstly obtained by solving a stochastic differential game problem with undetermined cost function. The cost function is then determined by the requirement of minimizing the maximal Lyapunov exponent of the controlled system in worst case.
在第二部分中,基于随机平均法和随机微分对策,提出了受有界的参数和外部扰动的拟Hamilton系统的极小极大最优控制策略。寻求最坏情况下最优控制的目标通过求解一个随机微分对策问题实现,而最坏情况扰动与相应的最优控制由Hamilton-Jacobi-Isaacs方程确定。作为上述方法的一个推广,详细阐述了其在反馈稳定化问题中的应用。通过求解一个具待定成本函数的随机微分对策问题确定系统的最坏情况扰动及相应的最优控制的形式,然后以最坏扰动下最优控制系统的最大Lyapunov指数最小为准则确定成本函数。
The present dissertation consists of two parts. In the first part, the robustness of the nonlinear stochastic optimal control for uncertain quasi Hamiltonian systems is studied. Based on the independence of uncertain parameters and stochastic excitations, the nonlinear stochastic optimal control for the nominal quasi Hamiltonian system with average-value parameters is firstly obtained by using the stochastic averaging method and stochastic dynamical programming principle. Then, the means and standard deviations of root-mean-square responses, control effectiveness and control efficiency for the uncertain quasi Hamiltonian system are calculated by using the stochastic averaging method and the probabilistic analysis. By introducing the sensitivities of the variation coefficients of controlled root-mean-square responses, control effectiveness and control efficiency to those of the uncertain parameters as measures, the robustness of the nonlinear stochastic optimal control is evaluated. As the application of the nonlinear stochastic optimal control strategy, controlled Bouc-Wen and Preisach hysteretic systems are studied and the robustness of the control strategy is also analyzed. Numerical results show the remarkable robustness of nonlinear stochastic optimal control.
In the second part, the robust control of uncertain quasi Hamiltonian systems is studied. A minimax optimal control strategy for quasi Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. The proposed strategy is searching for the optimal worst-case controller by solving a stochastic differential game problem, and the worst-case disturbances and the corresponding optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs equation. The feedback stabilization of uncertain quasi Hamiltonian systems are investigated as a generalization of the above robust control strategy. The form of the worst-case disturbances and the optimal controls are firstly obtained by solving a stochastic differential game problem with undetermined cost function. The cost function is then determined by the requirement of minimizing the maximal Lyapunov exponent of the controlled system in worst case.
引文
[1]Yong,J.M.,Zhou,X.Y.(1999),Stochastic Controls,Hamiltonian Systems and HJB Equations,Springer-Verlag,New York.
[2]Kovaleva,A.(1999).Optimal Control of Mechanical Oscillations,SpringerVerlag,Berlin.
[3]Pontryagin,L.S.,Boltyanski,V.G.,Garnkrelidze,R.V.,Mischenko,E.F.(1962),Mathematical Theory of Optimal Processes,Wiley,New York.
[4]Bellman,R.(1957),Dynamic Programming,Princeton University Press,Princeton.
[5]Crandall,M.G.,Lions,P.L.(1983),"Viscosity solutions of Hamilton-Jacobi equations;" Transaction of American Mathematics Society,277,1-42.
[6]陈滨.(1987),分析动力学,北京:北京大学出版社.
[7]Arnold,V.I.(1989),Mathematical Methods of Classical Mechanics,2nd edition,Springer-Verlag,New York.
[8]Arnold,V.I.,Kozlov,V.V.,Neishtadt,A.I.(1988),Mathematical Aspect of Classical and Celestial Mechanics,in Dynamical Systems Ⅲ,Arnold V I(Ed.),Springer-Verlag,Berlin.
[9]Ma,J.,Protter,P.,Yong,J.(1994),"Solving forward-backward stochastic differential equations explicitly-a four step scheme," Probability Theory and Related Fields,98,339-359.
[10]Hu,Y.,Peng,S.(1995),"Solution of forward-backward stochastic differential equations," Probability Theory and Related Fields,103,273-283.
[11]Yong,J.(1997),"Finding adapted solutions of forward-backward stochastic differential equations-method of continuation," Probability Theory and Related Fields,107,537-572.
[12]Ma,J.,Yong,J.(1999),Forward-Backward Stochastic Differential Equations and Their Applications,Springer-Verlag,Berlin.
[13]Kushner,H.J.(1967),Stochastic Stability and Control,Academic Press,New York.
[14]Fleming,W.H.,Rishel,R.W.(1975),Deterministic and Stochastic Optimal Control,Springer-Verlag,New York.
[15]Fleming,W.H.,Soner,H.M.(1993),Controlled Markov Processes and Viscosity Solutions,Springer-Verlag,New York.
[16]Kushner,H.J.,Dupuis,P.(1993),Numerical Methods for Stochastic Control Problems in Continuous Time,Springer-Verlag,New York.
[17]朱位秋.(2003),非线性随机动力学与控制—Hamilton理论体系框架,北京:科学出版社.
[18]Zhu,W.Q.,Yang,Y.Q.(1997),"Stochastic averaging of quasi-nonintegrable -Hamilton systems," ASME Journal of Applied Mechanics,64,157-164.
[19]Zhu,W.Q.,Huang,Z.L.,Yang,Y.Q.(1997),"Stochastic averaging of quasi integrable-Hamiltonian systems," ASME Journal of Applied Mechanics,64,975-984.
[20]Zhu,W.Q.,Huang,Z.L.,Suzuki,Y.(2002),"Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems," International Journal of Non-Linear Mechanics,37,419-437.
[21]Zhu,W.Q.,Ying,Z.G.(1999),"Optimal nonlinear feedback control of quasi-Hamiltonian systems," Science in China,Series A,42,1213-1219.
[22]Zhu,W.Q.,Ying,Z.G.,Soong,T.T.(2001),"An optimal nonlinear feedback control strategy for randomly excited structural systems," Nonlinear Dynamics,24,31-51.
[23]Zhu,W.Q.,Ying,Z.G.(2002),"Nonlinear stochastic optimal control of partially observable linear structures," Engineering Structures,24,333-342.
[24]Ying,Z.G.,Zhu,W.Q.(2008),"A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems," Journal of Sound and Vibration,310,184-196.
[25]Ying,Z.G.,Zhu,W.Q.,Soong,T.T.(2003),"A stochastic optimal semi-active control strategy for ER/MR dampers," Journal of Sound and Vibration,259,45-62.
[26]Dong,L.,Ying,Z.G.,Zhu,W.Q.(2004),"Stochastic optimal semi-active control of nonlinear systems using MR damper," Advances in Structural Engineering,7,485-494.
[27]Ying,Z.G.,Ni,Y.Q.,Ko,J.M.(2004),"A new stochastic optimal control strategy for hysteretic MR dampers," Acta Mechanica Solida Sinica,17,223-229.
[28]Cheng,H.,Zhu,W.Q.,Ying,Z.G.(2006),"Stochastic optimal semi-active control of hysteretic systems by using a magnetorheological damper," Smart Material &Structures,15,711-718.
[29]Ying,Z.G.,Ni,Y.Q.,Ko,J.M.(2007),"A bounded stochastic optimal semi-active control," Journal of Sound and Vibration, 304, 948-956.
[30] Zhu, W. Q., Deng, M. L. (2004), "Optimal bounded control for minimizing the response of quasi integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 39, 1535-1546.
[31] Zhu, W. Q., Deng, M. L. (2004), "Optimal bounded control for minimizing the response of quasi nonintegrable Hamiltonian systems," Nonlinear Dynamics, 35, 81-100.
[32] Ying, Z. G., Zhu, W. Q. (2006), "A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation," Automatica, 42, 1577-1582.
[33] Zhu, W. Q., Huang, Z. L. (2003), "Feedback stabilization of quasi-integrable- Hamiltonian systems," ASME Journal of Applied Mechanics, 70, 129-136.
[34] Zhu, W. Q., Huang, Z. L. (2003), "Stochastic stabilization of quasi partially integrable Hamiltonian systems by using Lyapunov exponent," Nonlinear Dynamics, 33, 209-224.
[35] Zhu, W. Q., Huang, Z. L. (2004), "Stochastic stabilization of quasi nonintegrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 39, 879-895.
[36] Zhu, W. Q. (2004), "Feedback stabilization of quasi nonintegrable Hamiltonian systems by using Lyapunov exponent," Nonlinear Dynamics, 36, 455-470.
[37] Zhu, W. Q., Huang, Z. L., Deng, M. L. (2002), "Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 37, 1057-1071.
[38] Zhu, W. Q., Huang, Z. L., Deng, M. L. (2003), "First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 38 (8), 1133-1148.
[39] Zhu, W. Q., Wu, Y. J. (2004), "Optimal bounded control of first-passage failure of strongly nonlinear oscillators under combined harmonic and white noise excitations," Journal of Sound and Vibration, 271(1), 83-101.
[40] Zhu, W. Q., Ying, Z. G, Ni, Y. Q., Ko, J. M. (2000), "Optimal nonlinear stochastic control of hysteretic systems," ASCE Journal of Engineering Mechanics, 126, 1027-1032.
[41] Ying, Z. G, Zhu, W. Q. (2003), "Stochastic optimal control of hysteretic systems under externally and parametrically random excitations," Acta Mechanica Solida Sinica, 16, 61-66.
[42] Ying, Z.G., Ni, Y.Q., Ko, J.M. (2004), "Non-linear stochastic optimal control for coupled-structures system of multi-degree-of-freedom," Journal of Sound and Vibration, 274, 843-861.
[43] Stengel, R. F., Ray, L. R., Marrison, C. I. (1995), "Probability evaluation of control system robustness," Int. J. Systems. Sci., 26, 1363-1382.
[44] Kozin, F. (1969), "A survey of stability of stochastic systems," Automatica, 5, 96-112.
[45] Yaz, E. (1988), "Deterministic and stochastic robustness measures for discrete systems," IEEE Transactions on Automatic Control, 33(10), 952-955.
[46] Miller, I., Freund, J. E. (1977), Probobility and Statistics for Engineers, Englewood Cliffs, NJ: Prentice Hall.
[47] Papoulis, A. (1990), Probobility and Statistics, Englewood Cliffs, NJ:Prentice Hall.
[48] Brown, G. W. (1956), Monte Carlo Method. Modern Mathmatics for Engineers, edit by E. F. Beckenbach, McGraw-Hill, New York.
[49] Apostolakis, G. (1990), "The concept of probability in safety assessments of technological systems," Science, 250, 1359-1364.
[50] Henley, E. J., Kumamoto, H. (1992), Probabilistic Risk Assessment: Reliability Engineering, Design and Analysis, IEEE Press, New York.
[51] Stengel, R. F. (1980), "Some effects, of parameter variations on the lateral-directional stability of aircraft," Journal of Guidance and Control, 3(2), 124-131.
[52] Stengel, R. F. (1994), Optimal Control and Estimation, Dover, New York.
[53] Stengel, R. F., Ray, L. R. (1991), "Stochastic robustness of linear-time-invariant control systems," IEEE Transactions on Automatic Control, 36(1), 82-87.
[54] Ray, L. R., Stengel, R. F. (1990), "Stochastic stability and performance robustness of linear multivariable systems," Proceedings of the America Control Conference, 462-467.
[55] Ray, L. R., Stengel, R. F. (1991), "Application of stochastic robustness to aircraft systems," Journal of Guidance, Control and Dynamics, 14(6), 1251-1259.
[56] Ray, L. R., Stengel, R. F. (1993), "Computer-aided analysis of linear control system robustness," Mechatronics, 3(1), 119-124.
[57] Ray, L. R., Stengel, R. F. (1993), "A Monte Carlo approach to the analysis of control system robustness," Automatica, 29, 229-236.
[58] Conover, W. J. (1980), Practical Non-parameter Statistics, Wiley, New York.
[59] Anderson, T. W., Burnstein, H. (1967), "Approximating the upper binomial confidence limt," Journal of the American Statistical Association, 62, 1413-1415.
[60] Marrison, C. I., Stengel, R. F. (1997), "Robust control system design using random search and genetic algorithms," IEEE Transactions on Automatic Control, 42, 835-839.
[61] Marrison, C. I., Stengel, R. F. (1998), "Design of robust systems for a hypersonic aircraft," Journal of Guidance, Control, and Dynamics, 21, 58-63.
[62] Wang, Q., Stengel, R. F. (2000), "Robust nonlinear control of a hypersonic aircraft," Journal of Guidance, Control, and Dynamics, 23, 577-585.
[63] Wang, Q., Stengel, R. F. (2001), "Searching for robust minimal-order compensators," Journal of Dynamic Systems, Measurement, and Control, 123, 233-235.
[64] Wang, Q., Stengel, R. F. (2002), "Robust control of nonlinear systems with parametric uncertainty," Automatica, 38,1591-1599.
[65] Motoda, T., Stengel, R. F., Miyazawa, Y. (2002), "Robust control system design using simulated annealing," Journal of Guidance, Control, and Dynamics, 25, 267-273.
[66] Weinmann, A. (1991), Uncertain Models and Robust Control, Springer-Verlag, New York.
[67] Zhou, K. M., Doyle, J. C, Glover, K. (1996), Robust and Optimal Control, Prentice-Hall, New Jersey.
[68] Zhou, K., Doyle, J. C. (1998), Essentials of Robust Control, Prentice Hall, New Jersey.
[69] Chen, B. M. (2000), Robust and H-Infinity Control, Springer, London.
[70] Petersen, I. R., James, M. R. (1996), "Performance analysis and controller synthesis for nonlinear systems with stochastic uncertainty constraints," Automatica, 32(7): 959-972.
[71] Petersen, I. R., McFarlane, D. C, Rotea, M. A. (1998), "Optima; guaranteed cost control of discrete-time uncertain linear systems," International Journal of Robust and Nonlinear Control, 8(8), 649-657.
[72] Petersen, I. R., Ugrinovskii, V. A., Savkin, A. V. (2000), Robust Control Design using H-infinity Methods, Springer-Verlag, London.
[73] Petersen, I. R., James, M. R., Dupuis, P. (1997), "Minimax optimal control of stochastic uncertain systems with relative entropy constraints," In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.
[74] Ugrinovskii, V. A., Petersen, I. R. (1997), "Finite horizon minimax optimal control of nonlinear continuous time systems with stochastic uncertainty," In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.
[75] Isaacs, R. (1965), Differential Game, Wiley, New York.
[76] Maitra, A. P., Sudderth, W. D. (1996), Discrete Gambling and Stochastic Games, Springer, New York.
[77] Tempo, R., Calafiore, G, Dabbene, F. (2005), Randomized Algorithms for Analysis and Control of Uncertain Systems, Springer-Verlag, London.
[78] Wen, Y. K. (1989), "Methods of random vibration for inelastic structures," Appl. Mech Rev., 42, 39-51.
[79] Wen, Y. K. (1993), "Inelastic structures under nonstationary random excitation," Nonlinear Dynamics and Stochastic Mechanics, W. Kliemann and N. Sri Namachchivaya, ed., CRC, Boca Raton, Fla., 411-435.
[80] Zhu, W. Q., Lei, Y. (1988), "Stochastic averaging of energy envelope of bilinear hysteretic system," Nonlinear Stochastic Dynamic Engineering Systems, F. Ziegler and G. I. Schueller, ed., Springer, Berlin, 381-391.
[81] Zhu, W. Q., Lin, Y. K. (1991), "Stochastic averaging of energy envelope," J. Engrg. Meck, ASCE, 117(8), 1890-1905.
[82] Roberts, J. B., Sadeghi, A. H. (1992), "Distribution of the response of hysteretic oscillators with wide-band random excitation," Proc., IUTAM Symp., Nonlinear Stochastic Meck, N. Bellomo and F. Casciati, ed., Springer, Berlin, 439-452.
[83] Cai, G. Q., Lin, Y. K. (1996), "Response of a hysteretic system under non-stationary earthquake excitations," Proc, IUTAM Symp., Adv. in Nonlinear Stochastic Mech., A. Naess and S. Krenk, ed., Kluwer Academic, Dordrecht, The Netherlands, 99-108.
[84] Housner, G. W., Bergman, L. A., Caughey, T. K., Chassiakos, A. G., Claus, R. O., Masri, S. F., Skelton, R. E., Soong, T. T., Spencer, B. F., Yao, J. T. P. (1997), "Structural control: Past, present, and future," J. Engrg. Mech, ASCE, 123 (9), 897-971.
[85] Yang, J. N., Wu, J. C, Agrawal, A. K. (1995), "Sliding mode control for nonlinear and hysteretic structures," J Engrg. Mech, ASCE, 121(12), 1330-1339.
[86] Yang, J. N., Agrawal, A. K., Chen, S. (1996), "Optimal polynomial control for seismically excited non-linear and hysteretic structures," Earthquake Engrg. and Struct. Dyn., 25, 1211-1230.
[87]Lin,Y.K.,Cai,G.Q.(1995),Probabilistic Structural Dynamics:Advanced Theory and Application,McGraw-Hill,New York.
[88]Wen,Y.K.(1976),"Method for random vibration of hysteretic systems," J.Engrg.Mech.Div.,ASCE,102(2),249-263.
[89]Krasnoselski,M.A.,Pokrovskii,A.V.(1989),Systems with Hysteresis,Springer,Berlin.
[90]Mayergoyz,I.D.(1991),Mathematical Models of Hysteresis,Springer,New York.
[91]Visintin,A.(1994),Differential Models of Hysteresis,Springer,Berlin.
[92]Spencer,B.F.,Dyke,S.J.,Sain,M.K.,Carlson,J.D.(1997),"Phenomenological model for magnetorheological dampers," J.Eng.Mech.,123,230-238.
[93]Hughes,D.,Wen,J.T.(1997),"Preisach modeling of piezoceramic and shape memory alloy hysteresis," Smart Mater.Struct.,6(3),287-300.
[94]Majima,S.,Kodama,K.,Hasegawa,T.(2001),"Modeling of shape memory alloy actuator and tracking control system with the model," IEEE Transactions on Control Systems Technology,9(1),54-59.
[95]Yu,Y.H.,Naganathan,N.,Dukkipati,R.(2002),"Preisach modeling of hysteresis for piezoceramic actuator system," Mechanism and Machine Theory,37,49-59.
[96]Caughey,T.K.(1960),"Random excitation of a system with bilinear hysteresis,"J.Appl.Mech.,27,649-652.
[97]Caughey,T.K.(1971),"Nonlinear theory of random vibrations," Adv.Appl.Mech.,11,209-253.
[98]Bouc,R.(1967),"Forced vibration of mechanical systems with hysteresis,"Abstract,Proc.Fourth Conf.on Nonlinear Oscillation,Prague,Czechoslovakia.
[99]Lubarda,V.,Sumarac,D.,Krajcinovic,D.(1993),"Hysteretic response of ductile materials," Eur.J.Mech.A/Solids,12(4),445-470.
[100]Preisach,F.(1935),"Uber die magnetische nachwirkung," Z.Phys.,94,277-302.
[101]Ying,Z.G.,Zhu,W.Q.,Ni,Y.Q.,Ko,J.M.(2002),"Stochastic averaging of Duhem hysteretic systems,"J.Sound Vib.,254(1),91-104.
[102]Mayergoyz,I.D.,Korman,C.E.(1991),"Preisach model with stochastic input as a model for viscosity," J.Appl.Phys.,69(4),2128-2134.
[103]Mayergoyz,I.D.,Korman,C.E.(1994),"The Preisach model with stochastic input as a model for after effect," J.Appl.Phys.,75(10),5478-5480.
[104]Korman,C.E.,Mayergoyz,I.D.(1997),"Review of Preisach-type models driven by stochastic inputs as a model for after effect," Physica B,233,381-389.
[105]Ni,Y.Q.,Ying,Z.G.,Ko,J.M.(2002),"Random response analysis of Preisach hysteretic systems with symmetric weight distribution," J. Appl. Mech., 69, 171-178.
[106] Ying, Z. G., Zhu, W. Q., Ni, Y. Q., Ko, J. M. (2002), "Random response of Preisach hysteretic systems," J. Sound Vib., 254(1), 37-49.
[107] Spanos, P. D., Cacciola, P., Muscolino, G. (2004), "Stochastic averaging of Preisach hysteretic systems," J. Eng Mech., 130(11), 1257-1267.
[108] Spanos, P. D., Cacciola, P., Redhorse, J. (2004), "Random vibration of SMA systems via Preisach formalism," Nonlinear Dynamics, 36, 405-419.
[109] Hughes, D., Wen, J. T. (1995), "Preisach modeling and compensation for smart material hysteresis," Active Materials and Smart Structures, Anderson GL, Lagoudas DC. eds. SPIE, Bellingham, WA, 2427, 50-64.
[110] Xu, Z., Chung, Y. K. (1994), "Averaging method using generalized harmonic functions for strongly non-linear oscillators," J. Sound Vib., 174, 563-576.
[111] Zhu, W. Q., Huang, Z. L., Suzuki, Y. (2001), "Response and stability of strongly non-linear oscillators under wide-band random excitation," International Journal of Non-linear Mechanics, 36, 1235-1250.
[112] Khasminskii, R. Z. (1980), Stochastic Stability of Differential Equation, Sijthoff &Noordhoff, Alphen aan den Rijn, The Netherlands.
[113] Afanasev, V. N., Kolmanovskii, V. B., Nosov, V. R. (1996), Mathematical Theory of Control Systems Design, Kluwer, Dordrecht, The Netherlands.
[114] Florchinger P. (1993), "A universal formula for the stabilization of control stochastic differential equations," Stochastic Analysis and Applications, 11, 155-162.
[115] Florchinger P. (1995), "Lyapunov-like techniques for stability," SIAM Journal of Control and Optimization, 33, 1151 -1169.
[116] Florchinger, D. (1997), "Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method," SIAM Journal of Control and Optimization, 35, 500-511.
[117] Krstic, M., Deng, H. (1998), Stabilization of Nonlinear Uncertain Systems, Springer, London.
[118] Oseledec, V. I. (1968), "A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems," Transactions of Moscow Mathematics Society, 19, 197-231.
[119] Khasminskii, R. Z. (1967), "Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems," Theory of Probability & Applications,12,144-147.
[120]Ariaratnam,S.T.Xie,W.C.(1992),"Lyapunov exponent and stochastic stability of coupled linear systems under real noise excitations," ASME Journal of Applied Mechanics,59,664-673.
[121]Zhu,W.Q.,Huang,Z.L.(1999),"Lyapunov exponents and stochastic stability of quasi-integrable-Hamiltonian systems," ASME Journal of Applied Mechanics,66,211-217.
[122]Stratonovich,R.L.(1963),Topics in the Theory of Random Noise,Vol.1,Gorden and Breach,New York.
[123]Khasminskii,R.Z.(1966),"A limit theorem for the solutions of differential equations with random right-hand sides," Theory of Probability and Applications,11,390-406.
[124]Blankenship,G.Papanicolaou,G.C.(1978),"Stability and control of stochastic systems with wide-band noise disturbances," SIAM Journal of Applied Mathematics,34,437-476.
[2]Kovaleva,A.(1999).Optimal Control of Mechanical Oscillations,SpringerVerlag,Berlin.
[3]Pontryagin,L.S.,Boltyanski,V.G.,Garnkrelidze,R.V.,Mischenko,E.F.(1962),Mathematical Theory of Optimal Processes,Wiley,New York.
[4]Bellman,R.(1957),Dynamic Programming,Princeton University Press,Princeton.
[5]Crandall,M.G.,Lions,P.L.(1983),"Viscosity solutions of Hamilton-Jacobi equations;" Transaction of American Mathematics Society,277,1-42.
[6]陈滨.(1987),分析动力学,北京:北京大学出版社.
[7]Arnold,V.I.(1989),Mathematical Methods of Classical Mechanics,2nd edition,Springer-Verlag,New York.
[8]Arnold,V.I.,Kozlov,V.V.,Neishtadt,A.I.(1988),Mathematical Aspect of Classical and Celestial Mechanics,in Dynamical Systems Ⅲ,Arnold V I(Ed.),Springer-Verlag,Berlin.
[9]Ma,J.,Protter,P.,Yong,J.(1994),"Solving forward-backward stochastic differential equations explicitly-a four step scheme," Probability Theory and Related Fields,98,339-359.
[10]Hu,Y.,Peng,S.(1995),"Solution of forward-backward stochastic differential equations," Probability Theory and Related Fields,103,273-283.
[11]Yong,J.(1997),"Finding adapted solutions of forward-backward stochastic differential equations-method of continuation," Probability Theory and Related Fields,107,537-572.
[12]Ma,J.,Yong,J.(1999),Forward-Backward Stochastic Differential Equations and Their Applications,Springer-Verlag,Berlin.
[13]Kushner,H.J.(1967),Stochastic Stability and Control,Academic Press,New York.
[14]Fleming,W.H.,Rishel,R.W.(1975),Deterministic and Stochastic Optimal Control,Springer-Verlag,New York.
[15]Fleming,W.H.,Soner,H.M.(1993),Controlled Markov Processes and Viscosity Solutions,Springer-Verlag,New York.
[16]Kushner,H.J.,Dupuis,P.(1993),Numerical Methods for Stochastic Control Problems in Continuous Time,Springer-Verlag,New York.
[17]朱位秋.(2003),非线性随机动力学与控制—Hamilton理论体系框架,北京:科学出版社.
[18]Zhu,W.Q.,Yang,Y.Q.(1997),"Stochastic averaging of quasi-nonintegrable -Hamilton systems," ASME Journal of Applied Mechanics,64,157-164.
[19]Zhu,W.Q.,Huang,Z.L.,Yang,Y.Q.(1997),"Stochastic averaging of quasi integrable-Hamiltonian systems," ASME Journal of Applied Mechanics,64,975-984.
[20]Zhu,W.Q.,Huang,Z.L.,Suzuki,Y.(2002),"Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems," International Journal of Non-Linear Mechanics,37,419-437.
[21]Zhu,W.Q.,Ying,Z.G.(1999),"Optimal nonlinear feedback control of quasi-Hamiltonian systems," Science in China,Series A,42,1213-1219.
[22]Zhu,W.Q.,Ying,Z.G.,Soong,T.T.(2001),"An optimal nonlinear feedback control strategy for randomly excited structural systems," Nonlinear Dynamics,24,31-51.
[23]Zhu,W.Q.,Ying,Z.G.(2002),"Nonlinear stochastic optimal control of partially observable linear structures," Engineering Structures,24,333-342.
[24]Ying,Z.G.,Zhu,W.Q.(2008),"A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems," Journal of Sound and Vibration,310,184-196.
[25]Ying,Z.G.,Zhu,W.Q.,Soong,T.T.(2003),"A stochastic optimal semi-active control strategy for ER/MR dampers," Journal of Sound and Vibration,259,45-62.
[26]Dong,L.,Ying,Z.G.,Zhu,W.Q.(2004),"Stochastic optimal semi-active control of nonlinear systems using MR damper," Advances in Structural Engineering,7,485-494.
[27]Ying,Z.G.,Ni,Y.Q.,Ko,J.M.(2004),"A new stochastic optimal control strategy for hysteretic MR dampers," Acta Mechanica Solida Sinica,17,223-229.
[28]Cheng,H.,Zhu,W.Q.,Ying,Z.G.(2006),"Stochastic optimal semi-active control of hysteretic systems by using a magnetorheological damper," Smart Material &Structures,15,711-718.
[29]Ying,Z.G.,Ni,Y.Q.,Ko,J.M.(2007),"A bounded stochastic optimal semi-active control," Journal of Sound and Vibration, 304, 948-956.
[30] Zhu, W. Q., Deng, M. L. (2004), "Optimal bounded control for minimizing the response of quasi integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 39, 1535-1546.
[31] Zhu, W. Q., Deng, M. L. (2004), "Optimal bounded control for minimizing the response of quasi nonintegrable Hamiltonian systems," Nonlinear Dynamics, 35, 81-100.
[32] Ying, Z. G., Zhu, W. Q. (2006), "A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation," Automatica, 42, 1577-1582.
[33] Zhu, W. Q., Huang, Z. L. (2003), "Feedback stabilization of quasi-integrable- Hamiltonian systems," ASME Journal of Applied Mechanics, 70, 129-136.
[34] Zhu, W. Q., Huang, Z. L. (2003), "Stochastic stabilization of quasi partially integrable Hamiltonian systems by using Lyapunov exponent," Nonlinear Dynamics, 33, 209-224.
[35] Zhu, W. Q., Huang, Z. L. (2004), "Stochastic stabilization of quasi nonintegrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 39, 879-895.
[36] Zhu, W. Q. (2004), "Feedback stabilization of quasi nonintegrable Hamiltonian systems by using Lyapunov exponent," Nonlinear Dynamics, 36, 455-470.
[37] Zhu, W. Q., Huang, Z. L., Deng, M. L. (2002), "Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 37, 1057-1071.
[38] Zhu, W. Q., Huang, Z. L., Deng, M. L. (2003), "First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems," International Journal of Non-Linear Mechanics, 38 (8), 1133-1148.
[39] Zhu, W. Q., Wu, Y. J. (2004), "Optimal bounded control of first-passage failure of strongly nonlinear oscillators under combined harmonic and white noise excitations," Journal of Sound and Vibration, 271(1), 83-101.
[40] Zhu, W. Q., Ying, Z. G, Ni, Y. Q., Ko, J. M. (2000), "Optimal nonlinear stochastic control of hysteretic systems," ASCE Journal of Engineering Mechanics, 126, 1027-1032.
[41] Ying, Z. G, Zhu, W. Q. (2003), "Stochastic optimal control of hysteretic systems under externally and parametrically random excitations," Acta Mechanica Solida Sinica, 16, 61-66.
[42] Ying, Z.G., Ni, Y.Q., Ko, J.M. (2004), "Non-linear stochastic optimal control for coupled-structures system of multi-degree-of-freedom," Journal of Sound and Vibration, 274, 843-861.
[43] Stengel, R. F., Ray, L. R., Marrison, C. I. (1995), "Probability evaluation of control system robustness," Int. J. Systems. Sci., 26, 1363-1382.
[44] Kozin, F. (1969), "A survey of stability of stochastic systems," Automatica, 5, 96-112.
[45] Yaz, E. (1988), "Deterministic and stochastic robustness measures for discrete systems," IEEE Transactions on Automatic Control, 33(10), 952-955.
[46] Miller, I., Freund, J. E. (1977), Probobility and Statistics for Engineers, Englewood Cliffs, NJ: Prentice Hall.
[47] Papoulis, A. (1990), Probobility and Statistics, Englewood Cliffs, NJ:Prentice Hall.
[48] Brown, G. W. (1956), Monte Carlo Method. Modern Mathmatics for Engineers, edit by E. F. Beckenbach, McGraw-Hill, New York.
[49] Apostolakis, G. (1990), "The concept of probability in safety assessments of technological systems," Science, 250, 1359-1364.
[50] Henley, E. J., Kumamoto, H. (1992), Probabilistic Risk Assessment: Reliability Engineering, Design and Analysis, IEEE Press, New York.
[51] Stengel, R. F. (1980), "Some effects, of parameter variations on the lateral-directional stability of aircraft," Journal of Guidance and Control, 3(2), 124-131.
[52] Stengel, R. F. (1994), Optimal Control and Estimation, Dover, New York.
[53] Stengel, R. F., Ray, L. R. (1991), "Stochastic robustness of linear-time-invariant control systems," IEEE Transactions on Automatic Control, 36(1), 82-87.
[54] Ray, L. R., Stengel, R. F. (1990), "Stochastic stability and performance robustness of linear multivariable systems," Proceedings of the America Control Conference, 462-467.
[55] Ray, L. R., Stengel, R. F. (1991), "Application of stochastic robustness to aircraft systems," Journal of Guidance, Control and Dynamics, 14(6), 1251-1259.
[56] Ray, L. R., Stengel, R. F. (1993), "Computer-aided analysis of linear control system robustness," Mechatronics, 3(1), 119-124.
[57] Ray, L. R., Stengel, R. F. (1993), "A Monte Carlo approach to the analysis of control system robustness," Automatica, 29, 229-236.
[58] Conover, W. J. (1980), Practical Non-parameter Statistics, Wiley, New York.
[59] Anderson, T. W., Burnstein, H. (1967), "Approximating the upper binomial confidence limt," Journal of the American Statistical Association, 62, 1413-1415.
[60] Marrison, C. I., Stengel, R. F. (1997), "Robust control system design using random search and genetic algorithms," IEEE Transactions on Automatic Control, 42, 835-839.
[61] Marrison, C. I., Stengel, R. F. (1998), "Design of robust systems for a hypersonic aircraft," Journal of Guidance, Control, and Dynamics, 21, 58-63.
[62] Wang, Q., Stengel, R. F. (2000), "Robust nonlinear control of a hypersonic aircraft," Journal of Guidance, Control, and Dynamics, 23, 577-585.
[63] Wang, Q., Stengel, R. F. (2001), "Searching for robust minimal-order compensators," Journal of Dynamic Systems, Measurement, and Control, 123, 233-235.
[64] Wang, Q., Stengel, R. F. (2002), "Robust control of nonlinear systems with parametric uncertainty," Automatica, 38,1591-1599.
[65] Motoda, T., Stengel, R. F., Miyazawa, Y. (2002), "Robust control system design using simulated annealing," Journal of Guidance, Control, and Dynamics, 25, 267-273.
[66] Weinmann, A. (1991), Uncertain Models and Robust Control, Springer-Verlag, New York.
[67] Zhou, K. M., Doyle, J. C, Glover, K. (1996), Robust and Optimal Control, Prentice-Hall, New Jersey.
[68] Zhou, K., Doyle, J. C. (1998), Essentials of Robust Control, Prentice Hall, New Jersey.
[69] Chen, B. M. (2000), Robust and H-Infinity Control, Springer, London.
[70] Petersen, I. R., James, M. R. (1996), "Performance analysis and controller synthesis for nonlinear systems with stochastic uncertainty constraints," Automatica, 32(7): 959-972.
[71] Petersen, I. R., McFarlane, D. C, Rotea, M. A. (1998), "Optima; guaranteed cost control of discrete-time uncertain linear systems," International Journal of Robust and Nonlinear Control, 8(8), 649-657.
[72] Petersen, I. R., Ugrinovskii, V. A., Savkin, A. V. (2000), Robust Control Design using H-infinity Methods, Springer-Verlag, London.
[73] Petersen, I. R., James, M. R., Dupuis, P. (1997), "Minimax optimal control of stochastic uncertain systems with relative entropy constraints," In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.
[74] Ugrinovskii, V. A., Petersen, I. R. (1997), "Finite horizon minimax optimal control of nonlinear continuous time systems with stochastic uncertainty," In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.
[75] Isaacs, R. (1965), Differential Game, Wiley, New York.
[76] Maitra, A. P., Sudderth, W. D. (1996), Discrete Gambling and Stochastic Games, Springer, New York.
[77] Tempo, R., Calafiore, G, Dabbene, F. (2005), Randomized Algorithms for Analysis and Control of Uncertain Systems, Springer-Verlag, London.
[78] Wen, Y. K. (1989), "Methods of random vibration for inelastic structures," Appl. Mech Rev., 42, 39-51.
[79] Wen, Y. K. (1993), "Inelastic structures under nonstationary random excitation," Nonlinear Dynamics and Stochastic Mechanics, W. Kliemann and N. Sri Namachchivaya, ed., CRC, Boca Raton, Fla., 411-435.
[80] Zhu, W. Q., Lei, Y. (1988), "Stochastic averaging of energy envelope of bilinear hysteretic system," Nonlinear Stochastic Dynamic Engineering Systems, F. Ziegler and G. I. Schueller, ed., Springer, Berlin, 381-391.
[81] Zhu, W. Q., Lin, Y. K. (1991), "Stochastic averaging of energy envelope," J. Engrg. Meck, ASCE, 117(8), 1890-1905.
[82] Roberts, J. B., Sadeghi, A. H. (1992), "Distribution of the response of hysteretic oscillators with wide-band random excitation," Proc., IUTAM Symp., Nonlinear Stochastic Meck, N. Bellomo and F. Casciati, ed., Springer, Berlin, 439-452.
[83] Cai, G. Q., Lin, Y. K. (1996), "Response of a hysteretic system under non-stationary earthquake excitations," Proc, IUTAM Symp., Adv. in Nonlinear Stochastic Mech., A. Naess and S. Krenk, ed., Kluwer Academic, Dordrecht, The Netherlands, 99-108.
[84] Housner, G. W., Bergman, L. A., Caughey, T. K., Chassiakos, A. G., Claus, R. O., Masri, S. F., Skelton, R. E., Soong, T. T., Spencer, B. F., Yao, J. T. P. (1997), "Structural control: Past, present, and future," J. Engrg. Mech, ASCE, 123 (9), 897-971.
[85] Yang, J. N., Wu, J. C, Agrawal, A. K. (1995), "Sliding mode control for nonlinear and hysteretic structures," J Engrg. Mech, ASCE, 121(12), 1330-1339.
[86] Yang, J. N., Agrawal, A. K., Chen, S. (1996), "Optimal polynomial control for seismically excited non-linear and hysteretic structures," Earthquake Engrg. and Struct. Dyn., 25, 1211-1230.
[87]Lin,Y.K.,Cai,G.Q.(1995),Probabilistic Structural Dynamics:Advanced Theory and Application,McGraw-Hill,New York.
[88]Wen,Y.K.(1976),"Method for random vibration of hysteretic systems," J.Engrg.Mech.Div.,ASCE,102(2),249-263.
[89]Krasnoselski,M.A.,Pokrovskii,A.V.(1989),Systems with Hysteresis,Springer,Berlin.
[90]Mayergoyz,I.D.(1991),Mathematical Models of Hysteresis,Springer,New York.
[91]Visintin,A.(1994),Differential Models of Hysteresis,Springer,Berlin.
[92]Spencer,B.F.,Dyke,S.J.,Sain,M.K.,Carlson,J.D.(1997),"Phenomenological model for magnetorheological dampers," J.Eng.Mech.,123,230-238.
[93]Hughes,D.,Wen,J.T.(1997),"Preisach modeling of piezoceramic and shape memory alloy hysteresis," Smart Mater.Struct.,6(3),287-300.
[94]Majima,S.,Kodama,K.,Hasegawa,T.(2001),"Modeling of shape memory alloy actuator and tracking control system with the model," IEEE Transactions on Control Systems Technology,9(1),54-59.
[95]Yu,Y.H.,Naganathan,N.,Dukkipati,R.(2002),"Preisach modeling of hysteresis for piezoceramic actuator system," Mechanism and Machine Theory,37,49-59.
[96]Caughey,T.K.(1960),"Random excitation of a system with bilinear hysteresis,"J.Appl.Mech.,27,649-652.
[97]Caughey,T.K.(1971),"Nonlinear theory of random vibrations," Adv.Appl.Mech.,11,209-253.
[98]Bouc,R.(1967),"Forced vibration of mechanical systems with hysteresis,"Abstract,Proc.Fourth Conf.on Nonlinear Oscillation,Prague,Czechoslovakia.
[99]Lubarda,V.,Sumarac,D.,Krajcinovic,D.(1993),"Hysteretic response of ductile materials," Eur.J.Mech.A/Solids,12(4),445-470.
[100]Preisach,F.(1935),"Uber die magnetische nachwirkung," Z.Phys.,94,277-302.
[101]Ying,Z.G.,Zhu,W.Q.,Ni,Y.Q.,Ko,J.M.(2002),"Stochastic averaging of Duhem hysteretic systems,"J.Sound Vib.,254(1),91-104.
[102]Mayergoyz,I.D.,Korman,C.E.(1991),"Preisach model with stochastic input as a model for viscosity," J.Appl.Phys.,69(4),2128-2134.
[103]Mayergoyz,I.D.,Korman,C.E.(1994),"The Preisach model with stochastic input as a model for after effect," J.Appl.Phys.,75(10),5478-5480.
[104]Korman,C.E.,Mayergoyz,I.D.(1997),"Review of Preisach-type models driven by stochastic inputs as a model for after effect," Physica B,233,381-389.
[105]Ni,Y.Q.,Ying,Z.G.,Ko,J.M.(2002),"Random response analysis of Preisach hysteretic systems with symmetric weight distribution," J. Appl. Mech., 69, 171-178.
[106] Ying, Z. G., Zhu, W. Q., Ni, Y. Q., Ko, J. M. (2002), "Random response of Preisach hysteretic systems," J. Sound Vib., 254(1), 37-49.
[107] Spanos, P. D., Cacciola, P., Muscolino, G. (2004), "Stochastic averaging of Preisach hysteretic systems," J. Eng Mech., 130(11), 1257-1267.
[108] Spanos, P. D., Cacciola, P., Redhorse, J. (2004), "Random vibration of SMA systems via Preisach formalism," Nonlinear Dynamics, 36, 405-419.
[109] Hughes, D., Wen, J. T. (1995), "Preisach modeling and compensation for smart material hysteresis," Active Materials and Smart Structures, Anderson GL, Lagoudas DC. eds. SPIE, Bellingham, WA, 2427, 50-64.
[110] Xu, Z., Chung, Y. K. (1994), "Averaging method using generalized harmonic functions for strongly non-linear oscillators," J. Sound Vib., 174, 563-576.
[111] Zhu, W. Q., Huang, Z. L., Suzuki, Y. (2001), "Response and stability of strongly non-linear oscillators under wide-band random excitation," International Journal of Non-linear Mechanics, 36, 1235-1250.
[112] Khasminskii, R. Z. (1980), Stochastic Stability of Differential Equation, Sijthoff &Noordhoff, Alphen aan den Rijn, The Netherlands.
[113] Afanasev, V. N., Kolmanovskii, V. B., Nosov, V. R. (1996), Mathematical Theory of Control Systems Design, Kluwer, Dordrecht, The Netherlands.
[114] Florchinger P. (1993), "A universal formula for the stabilization of control stochastic differential equations," Stochastic Analysis and Applications, 11, 155-162.
[115] Florchinger P. (1995), "Lyapunov-like techniques for stability," SIAM Journal of Control and Optimization, 33, 1151 -1169.
[116] Florchinger, D. (1997), "Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method," SIAM Journal of Control and Optimization, 35, 500-511.
[117] Krstic, M., Deng, H. (1998), Stabilization of Nonlinear Uncertain Systems, Springer, London.
[118] Oseledec, V. I. (1968), "A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems," Transactions of Moscow Mathematics Society, 19, 197-231.
[119] Khasminskii, R. Z. (1967), "Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems," Theory of Probability & Applications,12,144-147.
[120]Ariaratnam,S.T.Xie,W.C.(1992),"Lyapunov exponent and stochastic stability of coupled linear systems under real noise excitations," ASME Journal of Applied Mechanics,59,664-673.
[121]Zhu,W.Q.,Huang,Z.L.(1999),"Lyapunov exponents and stochastic stability of quasi-integrable-Hamiltonian systems," ASME Journal of Applied Mechanics,66,211-217.
[122]Stratonovich,R.L.(1963),Topics in the Theory of Random Noise,Vol.1,Gorden and Breach,New York.
[123]Khasminskii,R.Z.(1966),"A limit theorem for the solutions of differential equations with random right-hand sides," Theory of Probability and Applications,11,390-406.
[124]Blankenship,G.Papanicolaou,G.C.(1978),"Stability and control of stochastic systems with wide-band noise disturbances," SIAM Journal of Applied Mathematics,34,437-476.