应用MR阻尼器的滞迟系统的非线性随机最优半主动控制
摘要
第一章中，简单介绍了随机控制理论的发展概况和研究内容及振动控制分类，并结合滞迟系统的特性简单回顾了求解滞迟系统随机响应的各种方法，提出滞迟系统的半主动控制为本文研究内容。第二章中，介绍了磁流变阻尼器的工作原理、阻尼力模型以及控制策略，提出了将磁流变阻尼器应用于滞迟系统的随机最优半主动控制。第三章中，介绍能量包线随机平均法和随机最优控制策略。第四章中，基于第三章中给出的基本原理对滞迟系统进行半主动控制。首先将滞迟系统转化为等效非线性非滞迟系统，利用能量包线随机平均法推导出系统的部分平均It(?)随机微分方程。然后基于动态规划原理建立系统的动态规划方程，求解动态规划方程得到系统的最优控制策略。将最优控制律代入系统的部分平均It(?)随机微分方程并完成平均得到系统的完全平均It(?)随机微分方程和与之对应的FPK方程。最终，通过求解FPK方程和系统的动态规划方程得到系统的响应。同时，还采用基于等效线性化的LQG控制策略对该滞迟系统进行控制。结果表明，本文提出的随机最优半主动控制策略比LQG控制策略具有更好的控制效果和控制效率。
In chapter 1, the present status of the stochastic control theory is briefly reviewed, and many methods for predicting the stochastic response of the hysteretic system are introduced. In chapter 2, the working principle of the MR damper and its mathematical models are expatiated, and the control strategies for using MR dampers are enumerated. In chapter 3, the stochastic averaging method of energy envelope and stochastic optimal control strategy are introduced. In chapter 4, based on the basic principles described in chapter 3, MR damper is used to semi-actively control the hysteretic system. First, the system is converted into an equivalent non-hysteretic nonlinear stochastic system, from which a partially averaged It(?) stochastic equation is derived by using the stochastic averaging method of energy envelope. For the ergodic control problem, a dynamical programming equation is established based on the stochastic dynamical programming principle and solved to yield the optimal control law. The fully averaged It(?) equation is obtained by substituting the optimal control force into partially averaged It(?) equation and completing the average. Finally, the response of the controlled system is obtained by solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov equation associated with the fully averaged It(?) equation. Simultaneously, LQG control strategy is also used to control the same system. The numerical result indicates that both the effectiveness and efficiency of the former control strategy are better than those of the latter.
In chapter 1, the present status of the stochastic control theory is briefly reviewed, and many methods for predicting the stochastic response of the hysteretic system are introduced. In chapter 2, the working principle of the MR damper and its mathematical models are expatiated, and the control strategies for using MR dampers are enumerated. In chapter 3, the stochastic averaging method of energy envelope and stochastic optimal control strategy are introduced. In chapter 4, based on the basic principles described in chapter 3, MR damper is used to semi-actively control the hysteretic system. First, the system is converted into an equivalent non-hysteretic nonlinear stochastic system, from which a partially averaged It(?) stochastic equation is derived by using the stochastic averaging method of energy envelope. For the ergodic control problem, a dynamical programming equation is established based on the stochastic dynamical programming principle and solved to yield the optimal control law. The fully averaged It(?) equation is obtained by substituting the optimal control force into partially averaged It(?) equation and completing the average. Finally, the response of the controlled system is obtained by solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov equation associated with the fully averaged It(?) equation. Simultaneously, LQG control strategy is also used to control the same system. The numerical result indicates that both the effectiveness and efficiency of the former control strategy are better than those of the latter.
引文
1 王军，林家浩 剪切型非线性滞迟系统随机地震响应的虚拟激励分析，固体力学学报，2001，22(1)：23-30
2 刘豹 现代控制理论，机械工业出版社，2000
3 郭尚来 随机控制，清华大学出版社，2000
4 Canghey T K, Random excitation of a system with bilinear hysteresis. J. Appl. Mech., 27(1960), 649-652
5 Asano K and Iwan W D, An alternative approach to the random response of bylinear hysteresis systems. Earthquake Eng. Struct. Dyn., 12(1984), 229-236
6 Iwan W D, A distributed element model for hysteresis and its steady-state dynamic response, J. Appl. Mech., 33(1966), 893-900
7 Jennings P C, Earthquake response of a yielding structure, J. Eng. Mech., 91(1965), 41-68
8 Bouc R, Forced vibration of mechanical systems with hysteresis, proc. 4th conf. Nonlinear Oscillation, Prague, Czechoslovakia, 1967
9 Wen Y K, Method for random vibration of hysteretic systems, J. Eng. Mech., 102(1976), 249-263
10 Wen Y K, Equivalent linearization for hysteretic systems under random excitation, J. Eng. Mech., 47(1980), 150-154
11 Iyengar R N and Dash P K, Study of the random vibration of nonlinear systems by the Gaussian closure technique, J. Appl. Mech., 1978, 45(6), 393-399
12 佘锟，沈德安 滞回系统随机振动的累积量截断法，振动与冲击，1989， 42(2)：1-11
13 Roberts J B, The response of an oscillator with bilinear hysteresis to stationary random excitation, J. Appl. Mech., 45(1978), 923-928
14 Roberts J B, The yielding behavior of a randomly excited elastic-plastic structure, J. Sound Vib., 72(1980), 150-154
15 朱位秋，雷鹰 能量包线随机平均法在双线性迟滞系统随机响应分析中的应用，航空学报，1980 10(1)：28-34
1 Guangqiang Yang Large-scale magnetorheological fluid damper for vibration mitigation: modeling, testing and control. A doctor dissertation
2 宁欣，磁流变阻尼器减振控制技术的研究，沈阳建筑工程学院硕士学位论文，结构工程专业
3 丁立强，梅德庆，陈子辰，浦军 磁流变阻尼技术及其工程应用，机电工程，2003，20(5)：109-112
4 廖昌荣，汽车悬架系统磁流变阻尼器研究，重庆大学，2001
5 潘存治，杨绍普，申永军，磁流变阻尼器的一种机电耦合模型，第七届全国非线性动力学学术会议和第九届非线性振动学术议论文集
6 Brogan W L Modem control theory [M] Prentice-Hall Englewood Cliffs N.J 1991
7 Leitman G Semi-active control for vibration attenuation [J] J Intelligent Mat. Systems and Struct, 1994(5) 841-846
8 李秀领，李红男 磁流变阻尼器结构控制策略研究进展，防灾减灾工程学报，2004(9)：335-343
9 McClamroch N H closed loop structural control using electrorheological dampers. Proc., Am. Control Conf., American Automatic Control Council Washington D.C. 1995 4173-4177
10 Dyke S J, Spencer B F, Sain M K, et al. Experimental verification of semi-active structural control strategies using acceleration feedback. Proc., 3rd Int. Conf. on Motion and vibration, Japan society of Mechanical Engineering, Tokyo, 1996,(3): 291-296
11 Dyke S J, Spencer B F, Sain M K, et al. Semi-active response reduction using magnetorheological dampers. Proceeding of the IFAC world congress San Francisco, 1996.145-150
12 Dyke S J, Spencer B F, Sain M K, et al. Modeling and control of magnetorheological dampers for seismic response reduction. Smart material structure, 1996,(5):565-575
13 Jose A. Inaudi. Modulated homogeneous friction: a semi-active damping strategy. Earthquake engineering structure. Dyn. 1997,26:361-376
14 Jansen L M, Dyke S J. Semiactive control strategy for MR dampers: a comparative study. ASCE Journal of engineering mechanics, 2000,126: 795-803
15 Symans M D, Constantious M C. Semi-active control systems for seismic protection of structures: a state-of-the -art review. Engineering structures, 1999,21:469-487
16 Ying Z G, Zhu W Q and Soong T T 2003 A stochastic optimal control strategy for ER/MR dampers J. Sound Vib. 259(1) 45-62
17 Dong L, Ying Z G and Zhu W Q 2005 semi-active stochastic optimal control of nonlinear systems by using MR dampers Adv. Struct. Engrg. 7 485-494
1 朱位秋，随机振动，北京科学技术出版社，1992，1998
2 朱位秋，非线性随机动力学与控制，北京科学出版社，2003
3 Stratonovich R L, Topics in the theory of random noise, Gordon and Breach, Vol.1,1963; Vol.2,1967
4 Khasmiskii R Z, A limit theorem for the solutions of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), 390-405
5 Papanicolaou G C and Kohler W, Asymptotic theory of mixing stochastic ordinary differential equations, Commun. Pure Appl. Math., 27(1974), 641-668
6 Zhu W Q, On the method of stochastic averaging of energy envelope, proc. Int. Workshop on Stochastic Struc. Mech., Rep 1-83, University of Innsbruck, 1983
7 Khasminskii R Z, On the averaging principle for stochastic differential It?? equations, Kibemetika, 4(1968), 160-179
8 Roberts J B, The energy envelope of a randomly excited nonlinear oscillator, J. Souns Vib., 60(1978), 177-185
9 Spanos P D, A closed form solution for a class of nonstationary nonlinear random vibration problem
10 Zhu W Q, Lin Y K, Stochastic averaging pf energy envelope, J. Eng. Mech., 117(1991), 1890-1905
11 Pontryagin L S, Boltyanski V G, Gamkrelidze R V, Mischenko E F. Mathematical Theory of Optimal Processes. New York: Wiley, 1962
12 Bellman R. Dynamic Programming. Princeton: Princeton University Press, 1957
13 Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Transaction of American Mathematics Society, 1983, 277:1-42
14 Kushner H J. Stochastic Stability and Control. New York: Academic Press, 1967
15 Fleming W H, Rishel R W. Deterministic and Stochastic Optimal Control. New York: Springer-Verlag, 1975
16 Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions. New York: Springer-Verlag, 1992
17 Kushner H J, Dupuis P. Numerical Methods for Stochastic Control Problems in Continuous Time, New York: Springer-Verlag, 1993
1 Yang J N, Wu J C and Agrawal A K, Sliding mode control for nonlinear and hysteretic structures J. Engrg. Mech. 121 (1995) 1330-1339
2 Roberts J B, Application of averaging methods to randomly excited hysteresis systems In: Ziegler F and Schu-??ller G I ed, Nonlinear Stochastic Dynamic En gineering Systems, Springer-Verlag, 1988
3 Booton R C, The analysis of nonlinear control systems with random inputs, IRE Trans. Cirruit Theory, 1(1995),32-34
4 Caughey T K, Equivalent linearization techniques, J. Acoust. Soc. Am., 85(1963), 1706-1711
5 Iwan W D, Yang I, Application of statistical linearization technique to nonlinear multi-degree-of-freedom systems, J. Appl. Mech.,39(1972),545-550
6 Iwan W D, A generalization of the method of equivalent linearization, J. Non-linear Mech.,8(1973),279-287
7 Spanos P D, Iwan W D, On the existence and uniqueness of solutions generated by equivalent linearization. J. Non-linear Mech., 13 (1978),71-78
8 Iwan W D, Application of nonlinear analysis techniques, Appl. Mech. Earthquake Engrg., 8(1974), 135-162
9 Iwan W D, Mason A B, Equivalent linearization for systems subjected to non-stationary random excitation, J. Non-linear Mech., 15(1980),71-82
10 Atalik, Utku S, Stochastic linearization of multi-degree-of-freedom nonlinear systems, Earthquake Eng. Strut. Dyn., 4(1976),411-420
11 Lin Y K, Cai Q G (1995) Probabilistic structural dynamics: advanced theory and application, McGraw-Hill, New York
12 Zhu W Q, Ying Z G, Ni Y Q, Ko J M Optimal nonlinear stochastic control of hysteretic systems ASCE Journal of engineering mechanics 2000,126: 1027-1032
13 朱位秋 随机振动 北京科学出版社 1992,1998
14 Zhu W Q, Lin Y K (1991) Stochastic averaging of energy envelope. Journal of engineering Journal of engineering mechanics. ASCE, 117(8), 1890-1895
15 Pontryagin L S, Boltyanski V G, Gamkrelidze R V, Miscjenko E F. Mathematical theory of optimal processes. New York: Wiley, 1962
16 Bellman R. Dynamical programming. Princeton: Princeton university press, 1957
17 Crandall M G, Lions P L, Viscosity solutions of Hamilton-Jacobi equations. Transaction of American Mathematics Society. 1983,277:1-42
18 Yang J N, Agrawal A K, Chen S, Optimal polynomial control for seismically excited non-linear and hysteretic structures. Earthquake engineering and structural dynamics. 25,1211-123
19 Roberts J B, The energy envelope of a randomly excited nonlinear oscillator, J. Souns Vib., 60(1978), 177-185
20 Zhu W Q, On the method of stochastic averaging of energy envelope, proc. Int. Workshop on Stochastic Struc. Mech., Rep 1-83, University of Innsbruck, 1983
21 Spanos P D, A closed form solution for a class of nonstationary nonlinear random vibration problem
22 Zhu W Q, Lin Y K, Stochastic averaging pf energy envelope, J. Eng. Mech., 117(1991), 1890-1905
23 Khasminskii R Z, On the averaging principle for stochastic differential Ito equations, Kibernetika, 4(1968), 260-179
24 Stratonovich R L, Topics in the theory of random noise. Gordon and Breach, vol. 1,1963; vol.2,1967
25 Fleming W H and Soner H M 1993 Controlled Markov processes and viscosity solutions (New York: Springer)
26 Sun L and Goto Y 1994 Applications of fuzzy theory to variable damper for bridge vibration control Proceeding of First World Conference on Structural Control WPI 31-40
27 Wen Y K 1989 Methods of random vibration for inelastic structures Appl. Mech. Rev.42 39-51
28 Wen Y K 1993 Inelastic structures under nonstationary random excitations Nonlinear dynamics and stochastic mechanics 411-435
29 Dong L, Ying Z G and Zhu W Q 2005 semi-active stochastic optimal control of nonlinear systems by using MR dampers Adv. Struct. Engrg. 7 485-494
2 刘豹 现代控制理论，机械工业出版社，2000
3 郭尚来 随机控制，清华大学出版社，2000
4 Canghey T K, Random excitation of a system with bilinear hysteresis. J. Appl. Mech., 27(1960), 649-652
5 Asano K and Iwan W D, An alternative approach to the random response of bylinear hysteresis systems. Earthquake Eng. Struct. Dyn., 12(1984), 229-236
6 Iwan W D, A distributed element model for hysteresis and its steady-state dynamic response, J. Appl. Mech., 33(1966), 893-900
7 Jennings P C, Earthquake response of a yielding structure, J. Eng. Mech., 91(1965), 41-68
8 Bouc R, Forced vibration of mechanical systems with hysteresis, proc. 4th conf. Nonlinear Oscillation, Prague, Czechoslovakia, 1967
9 Wen Y K, Method for random vibration of hysteretic systems, J. Eng. Mech., 102(1976), 249-263
10 Wen Y K, Equivalent linearization for hysteretic systems under random excitation, J. Eng. Mech., 47(1980), 150-154
11 Iyengar R N and Dash P K, Study of the random vibration of nonlinear systems by the Gaussian closure technique, J. Appl. Mech., 1978, 45(6), 393-399
12 佘锟，沈德安 滞回系统随机振动的累积量截断法，振动与冲击，1989， 42(2)：1-11
13 Roberts J B, The response of an oscillator with bilinear hysteresis to stationary random excitation, J. Appl. Mech., 45(1978), 923-928
14 Roberts J B, The yielding behavior of a randomly excited elastic-plastic structure, J. Sound Vib., 72(1980), 150-154
15 朱位秋，雷鹰 能量包线随机平均法在双线性迟滞系统随机响应分析中的应用，航空学报，1980 10(1)：28-34
1 Guangqiang Yang Large-scale magnetorheological fluid damper for vibration mitigation: modeling, testing and control. A doctor dissertation
2 宁欣，磁流变阻尼器减振控制技术的研究，沈阳建筑工程学院硕士学位论文，结构工程专业
3 丁立强，梅德庆，陈子辰，浦军 磁流变阻尼技术及其工程应用，机电工程，2003，20(5)：109-112
4 廖昌荣，汽车悬架系统磁流变阻尼器研究，重庆大学，2001
5 潘存治，杨绍普，申永军，磁流变阻尼器的一种机电耦合模型，第七届全国非线性动力学学术会议和第九届非线性振动学术议论文集
6 Brogan W L Modem control theory [M] Prentice-Hall Englewood Cliffs N.J 1991
7 Leitman G Semi-active control for vibration attenuation [J] J Intelligent Mat. Systems and Struct, 1994(5) 841-846
8 李秀领，李红男 磁流变阻尼器结构控制策略研究进展，防灾减灾工程学报，2004(9)：335-343
9 McClamroch N H closed loop structural control using electrorheological dampers. Proc., Am. Control Conf., American Automatic Control Council Washington D.C. 1995 4173-4177
10 Dyke S J, Spencer B F, Sain M K, et al. Experimental verification of semi-active structural control strategies using acceleration feedback. Proc., 3rd Int. Conf. on Motion and vibration, Japan society of Mechanical Engineering, Tokyo, 1996,(3): 291-296
11 Dyke S J, Spencer B F, Sain M K, et al. Semi-active response reduction using magnetorheological dampers. Proceeding of the IFAC world congress San Francisco, 1996.145-150
12 Dyke S J, Spencer B F, Sain M K, et al. Modeling and control of magnetorheological dampers for seismic response reduction. Smart material structure, 1996,(5):565-575
13 Jose A. Inaudi. Modulated homogeneous friction: a semi-active damping strategy. Earthquake engineering structure. Dyn. 1997,26:361-376
14 Jansen L M, Dyke S J. Semiactive control strategy for MR dampers: a comparative study. ASCE Journal of engineering mechanics, 2000,126: 795-803
15 Symans M D, Constantious M C. Semi-active control systems for seismic protection of structures: a state-of-the -art review. Engineering structures, 1999,21:469-487
16 Ying Z G, Zhu W Q and Soong T T 2003 A stochastic optimal control strategy for ER/MR dampers J. Sound Vib. 259(1) 45-62
17 Dong L, Ying Z G and Zhu W Q 2005 semi-active stochastic optimal control of nonlinear systems by using MR dampers Adv. Struct. Engrg. 7 485-494
1 朱位秋，随机振动，北京科学技术出版社，1992，1998
2 朱位秋，非线性随机动力学与控制，北京科学出版社，2003
3 Stratonovich R L, Topics in the theory of random noise, Gordon and Breach, Vol.1,1963; Vol.2,1967
4 Khasmiskii R Z, A limit theorem for the solutions of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), 390-405
5 Papanicolaou G C and Kohler W, Asymptotic theory of mixing stochastic ordinary differential equations, Commun. Pure Appl. Math., 27(1974), 641-668
6 Zhu W Q, On the method of stochastic averaging of energy envelope, proc. Int. Workshop on Stochastic Struc. Mech., Rep 1-83, University of Innsbruck, 1983
7 Khasminskii R Z, On the averaging principle for stochastic differential It?? equations, Kibemetika, 4(1968), 160-179
8 Roberts J B, The energy envelope of a randomly excited nonlinear oscillator, J. Souns Vib., 60(1978), 177-185
9 Spanos P D, A closed form solution for a class of nonstationary nonlinear random vibration problem
10 Zhu W Q, Lin Y K, Stochastic averaging pf energy envelope, J. Eng. Mech., 117(1991), 1890-1905
11 Pontryagin L S, Boltyanski V G, Gamkrelidze R V, Mischenko E F. Mathematical Theory of Optimal Processes. New York: Wiley, 1962
12 Bellman R. Dynamic Programming. Princeton: Princeton University Press, 1957
13 Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Transaction of American Mathematics Society, 1983, 277:1-42
14 Kushner H J. Stochastic Stability and Control. New York: Academic Press, 1967
15 Fleming W H, Rishel R W. Deterministic and Stochastic Optimal Control. New York: Springer-Verlag, 1975
16 Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions. New York: Springer-Verlag, 1992
17 Kushner H J, Dupuis P. Numerical Methods for Stochastic Control Problems in Continuous Time, New York: Springer-Verlag, 1993
1 Yang J N, Wu J C and Agrawal A K, Sliding mode control for nonlinear and hysteretic structures J. Engrg. Mech. 121 (1995) 1330-1339
2 Roberts J B, Application of averaging methods to randomly excited hysteresis systems In: Ziegler F and Schu-??ller G I ed, Nonlinear Stochastic Dynamic En gineering Systems, Springer-Verlag, 1988
3 Booton R C, The analysis of nonlinear control systems with random inputs, IRE Trans. Cirruit Theory, 1(1995),32-34
4 Caughey T K, Equivalent linearization techniques, J. Acoust. Soc. Am., 85(1963), 1706-1711
5 Iwan W D, Yang I, Application of statistical linearization technique to nonlinear multi-degree-of-freedom systems, J. Appl. Mech.,39(1972),545-550
6 Iwan W D, A generalization of the method of equivalent linearization, J. Non-linear Mech.,8(1973),279-287
7 Spanos P D, Iwan W D, On the existence and uniqueness of solutions generated by equivalent linearization. J. Non-linear Mech., 13 (1978),71-78
8 Iwan W D, Application of nonlinear analysis techniques, Appl. Mech. Earthquake Engrg., 8(1974), 135-162
9 Iwan W D, Mason A B, Equivalent linearization for systems subjected to non-stationary random excitation, J. Non-linear Mech., 15(1980),71-82
10 Atalik, Utku S, Stochastic linearization of multi-degree-of-freedom nonlinear systems, Earthquake Eng. Strut. Dyn., 4(1976),411-420
11 Lin Y K, Cai Q G (1995) Probabilistic structural dynamics: advanced theory and application, McGraw-Hill, New York
12 Zhu W Q, Ying Z G, Ni Y Q, Ko J M Optimal nonlinear stochastic control of hysteretic systems ASCE Journal of engineering mechanics 2000,126: 1027-1032
13 朱位秋 随机振动 北京科学出版社 1992,1998
14 Zhu W Q, Lin Y K (1991) Stochastic averaging of energy envelope. Journal of engineering Journal of engineering mechanics. ASCE, 117(8), 1890-1895
15 Pontryagin L S, Boltyanski V G, Gamkrelidze R V, Miscjenko E F. Mathematical theory of optimal processes. New York: Wiley, 1962
16 Bellman R. Dynamical programming. Princeton: Princeton university press, 1957
17 Crandall M G, Lions P L, Viscosity solutions of Hamilton-Jacobi equations. Transaction of American Mathematics Society. 1983,277:1-42
18 Yang J N, Agrawal A K, Chen S, Optimal polynomial control for seismically excited non-linear and hysteretic structures. Earthquake engineering and structural dynamics. 25,1211-123
19 Roberts J B, The energy envelope of a randomly excited nonlinear oscillator, J. Souns Vib., 60(1978), 177-185
20 Zhu W Q, On the method of stochastic averaging of energy envelope, proc. Int. Workshop on Stochastic Struc. Mech., Rep 1-83, University of Innsbruck, 1983
21 Spanos P D, A closed form solution for a class of nonstationary nonlinear random vibration problem
22 Zhu W Q, Lin Y K, Stochastic averaging pf energy envelope, J. Eng. Mech., 117(1991), 1890-1905
23 Khasminskii R Z, On the averaging principle for stochastic differential Ito equations, Kibernetika, 4(1968), 260-179
24 Stratonovich R L, Topics in the theory of random noise. Gordon and Breach, vol. 1,1963; vol.2,1967
25 Fleming W H and Soner H M 1993 Controlled Markov processes and viscosity solutions (New York: Springer)
26 Sun L and Goto Y 1994 Applications of fuzzy theory to variable damper for bridge vibration control Proceeding of First World Conference on Structural Control WPI 31-40
27 Wen Y K 1989 Methods of random vibration for inelastic structures Appl. Mech. Rev.42 39-51
28 Wen Y K 1993 Inelastic structures under nonstationary random excitations Nonlinear dynamics and stochastic mechanics 411-435
29 Dong L, Ying Z G and Zhu W Q 2005 semi-active stochastic optimal control of nonlinear systems by using MR dampers Adv. Struct. Engrg. 7 485-494