几类非线性随机系统动力学与控制研究
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摘要
第二章中,利用广义Hamilton系统的可积性与共振性,将广义Hamilton系统分为不可积、完全可积非共振、完全可积共振、部分可积非共振、部分可积共振五类。第三章中,得到了高斯白噪声激励下耗散的五类广义Hamilton系统的精确平稳解的泛函形式及解存在条件。第四章中,应用上述精确平稳解与三种等效准则(给定系统与等效系统阻尼力差的均方值最小的准则,给定系统与等效系统阻尼力耗散能量差的均方值最小准则及给定系统与等效系统首次积分的时间变化率的期望相等准则),提出了高斯白噪声激励下耗散的五类广义Hamilton系统的等效非线性系统法。第五章中,建立了高斯白噪声激励下耗散的拟不可积、拟完全可积非共振、拟完全可积共振、拟部分可积非共振、拟部分可积共振广义Hamilton系统的随机平均法,指出了平均方程的维数与可积性及共振性之间的关系,给出了平均方程漂移与扩散系数的求法。建立分别在谐和与高斯白噪声、有界噪声、平稳宽带随机激励下拟完全可积Hamilton系统的随机平均法及谐和激励下拟可积Hamilton系统的确定性平均法,并将该随机平均法应用于预测高斯白噪声激励的多自由度碰撞振动系统及平稳宽带随机激励下单自由度碰撞振动系统的平稳响应。第六章中,基于拟不可积广义Hamilton系统的随机平均法,引入Hamilton函数与Casimir函数之和的平方根新模,用最大Lyapunov指数研究了拟不可积广义Hamilton的概率为1渐近稳定性;利用Lyapunov指数对不同模定义的不变性,提出了确定高维线性随机系统几乎肯定渐近稳定域的近似方法;提出了利用独立运动积分的线性组合构造新Lyapunov函数的新方法,并用Lyapunov函数得到了随机激励下耗散的拟不可积、拟可积Hamilton系统概率为1渐近稳定性的充分条件;提出了研究拟可积Hamilton系统随机Hopf分岔的新近似方法;还研究了随机参数扰动对Lorenz系统族的影响;应用拟不可积Hamilton系统随机平均法,研究了参数随机扰动的最简单电力系统——单机无穷大系统的首次穿越时间。第七章中,利用拟Hamilton系统随机平均法及随机动态规划原理,提出了基于奇异边界类别或最大Lyapunov指数的反馈稳定化的方法;提出了以系统可靠度最大或平均首次穿越时间最长为目标的非线性随机最优控制策略,指出其有界遍历最优控制即bang-bang控制;指出了以最大可靠度
In Chapter 2, the generalized Hamiltonian systems (GHS) are classified as five groups based on their integrability and resonance, i.e., completely non-integrable, completely integrable and non-resonant, completely integrable and resonant, partially integrable and non-resonant, partially integrable and resonant. In Chapter 3, the functional form of the exact stationary solutions and the conditions for exact stationary solutions to exist for five groups of dissipated GHS under Gaussian white noise excitations are given. In Chapter 4, the equivalent nonlinear system methods for dissipated five classes of GHS under Gaussian white noise excitations are developed. Three equivalence criteria were proposed for finding the equivalent nonlinear system and their stationary solutions. They are minimization of the mean square deficiency in damping forces, minimization of the mean square deficiency in the energies dissipated by the damping forces, and equality of the expected time rates of the first integrals of the given system and its equivalent one. In Chapter 5, stochastic averaging methods for five classes of GHS under light damping and Gaussian white noise excitations are developed. The form and dimension of the averaged Ito stochastic differential equation depends upon the integrability and resonance of the associated GHS. The diffusion and drift coefficients of the averaged Ito equation are given in detail. The stochastic averaging methods for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations, under bounded noise excitations, and under stationary wide-band noise excitations, respectively, are also developed. The deterministic averaging method for completely integrable Hamiltonian system under light damping and weak harmonic excitations is the special case of the stochastic averaging method for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations. The developed stochastic averaging methods are used to predict the responses of single-degree-of-freedom vibro-impact system under light damping and stationary wide-band noise excitations, and the responses of multi-degree-of-freedom vibro-impact systems under light damping and Gaussian white noise excitations. In Chapter 6, the largest Lyapunov exponent and almost sure asymptotic stability of quasi non-integrable GHS under light damping and weak Gaussian white noise excitations are studied based on the stochastic averaging methods for quasi-non-integrable GHS and the definition of new norm in terms of square root of summation of Hamiltonian and Casimir functions. A new approach to the almost sure asymptotic stability of linear stochastic system is proposed by using the property that the largest Lyapunov exponent is independent of the positive-definite quadratic form in the definition of norm. The Lyapunov asymptotic stability with probability one of quasi Hamilton systems are studied by using Lyapunov function and the stochastic averaging methods for quasi Hamilton systems. The Lyapunov functions are the optimal linear combination of the first integrals of the associated completely integrable and non-resonant or partially integrable and non-resonant Hamiltonian systems. An approximate method for determing the stochastic Hopf bifurcation of quasi integrable Hamiltonian system is developed based on the stochastic averaging
In Chapter 2, the generalized Hamiltonian systems (GHS) are classified as five groups based on their integrability and resonance, i.e., completely non-integrable, completely integrable and non-resonant, completely integrable and resonant, partially integrable and non-resonant, partially integrable and resonant. In Chapter 3, the functional form of the exact stationary solutions and the conditions for exact stationary solutions to exist for five groups of dissipated GHS under Gaussian white noise excitations are given. In Chapter 4, the equivalent nonlinear system methods for dissipated five classes of GHS under Gaussian white noise excitations are developed. Three equivalence criteria were proposed for finding the equivalent nonlinear system and their stationary solutions. They are minimization of the mean square deficiency in damping forces, minimization of the mean square deficiency in the energies dissipated by the damping forces, and equality of the expected time rates of the first integrals of the given system and its equivalent one. In Chapter 5, stochastic averaging methods for five classes of GHS under light damping and Gaussian white noise excitations are developed. The form and dimension of the averaged Ito stochastic differential equation depends upon the integrability and resonance of the associated GHS. The diffusion and drift coefficients of the averaged Ito equation are given in detail. The stochastic averaging methods for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations, under bounded noise excitations, and under stationary wide-band noise excitations, respectively, are also developed. The deterministic averaging method for completely integrable Hamiltonian system under light damping and weak harmonic excitations is the special case of the stochastic averaging method for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations. The developed stochastic averaging methods are used to predict the responses of single-degree-of-freedom vibro-impact system under light damping and stationary wide-band noise excitations, and the responses of multi-degree-of-freedom vibro-impact systems under light damping and Gaussian white noise excitations. In Chapter 6, the largest Lyapunov exponent and almost sure asymptotic stability of quasi non-integrable GHS under light damping and weak Gaussian white noise excitations are studied based on the stochastic averaging methods for quasi-non-integrable GHS and the definition of new norm in terms of square root of summation of Hamiltonian and Casimir functions. A new approach to the almost sure asymptotic stability of linear stochastic system is proposed by using the property that the largest Lyapunov exponent is independent of the positive-definite quadratic form in the definition of norm. The Lyapunov asymptotic stability with probability one of quasi Hamilton systems are studied by using Lyapunov function and the stochastic averaging methods for quasi Hamilton systems. The Lyapunov functions are the optimal linear combination of the first integrals of the associated completely integrable and non-resonant or partially integrable and non-resonant Hamiltonian systems. An approximate method for determing the stochastic Hopf bifurcation of quasi integrable Hamiltonian system is developed based on the stochastic averaging
引文
[1] Crandall, S. H. , ed. , Random Vibration, The MIT Press, 1958
[2] Symposium on the Response of Nonliear Systems to Random Excitation, J. Acoust. Am. , 1963, 35: 1683—1721.
[3] Crandall, S. H. , and Mark, W. D. , Random Vibration in Mechanical Systems, Acadcmic Press, 1963.
[4] Robson, J. , D. , An Introduction to Random Vibration, Edinburgh University press, 1963. (中译本:J.D.罗伯逊,随机振动引论,湖南科学技术出版社,1980)
[5] Bolotin, V. V. , Statistical Methods in Stuctural Mechanics, Holden-Day, 1969.
[6] Newland, D. E, An Introduction to Random Vibration and Spectral Analysis, Longmans, 1stedit, 1975; 2nd edit, 1984. (第一版本中译本:D.E纽兰,随机振动与谱分析概论,机械工业版社,1980).
[7] Lin, Y. K. , Probabilistic Theory of Structural Dynamics, Mcgraw-Hill, 1967, (Repint by Robert E Krieger publishing Co, 1976).
[8] Nigam, N. C. , Introduction to Random Vibrations, The MIT press, 1983, (中译本:N.C.尼格姆,随机振动概论,上海交通大学出版社,1985).
[9] Crandall, S. H. , ed. , Random vibration, The MIT Press, 1963.
[10] Crandall S. H. and Zhu W. Q. Random Vibration: a survey of recent developments, ASME J. Appl. Mech. 50, 1983, 50th Anniversary Issue, 953-962.
[11] Spanos, P. D. , Stochastic linearization in Structural Dynamics, Appl. Mech. Rev. , 1981, 34: 1-8.
[12] Roberts, J. B. , and Spanos, R D. , Stochastic averaging: an approximate method of solving random vibration problems, Int. J, Non-Linear Mech. , 1986, 21: 111-134.
[13] Zhu, W. Q. , Stochassic averaging methods in random vibration, Appl. Mech. Rev. , 1988, 41 (5) : 189-199.
[14] Lin, Y. K. , and Cai, G. Q. , Vibration of nonlinear systems under additive and multiplicative random excitations, Appl. Mech. Rev. , 1989, 42 (11) : S135-S141.
[15] Bolotin V. V. , Random Vibration of Elastic Systems, Matinus Nijhoff Publishers, The Hague. 1984.
[16] Ibrahim R. A. , Parametric Random Vibration, Taunton: Reserch Studies Press LTD, 1985.
[17] Dimentberg M. F. , Statistical Dynamics of Nonlinear and Time-Varying Systems, Taunton: Reserch Sdudies Press LtD, England. , 1988.
[18] Roberts J. B. and Spanos P. D. , Random Vibration and Stctistical Linearization, New York: Wiley, 1990.
[19] Soong T. T. and Grigoriu M. , Random Vibration of Mechanical and Structural Systems, New Jersey: PTR Prentice-Hall, (1993),
[20] Soize C. , The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solution, Singapore: World Scientific, 1994.
[21] Lin Y. K. and Cai G. Q. , Probabilistic Structural Dynamics, Advanced Theory and Applications, New York: McGraw-Hill, 1995.
[22] Grigoriu M. , Applied Non-Gaussian Processes; Examples, Theory, Simulations, Linear Random Vibration and MATLAB Solutions, Englewood Cliffs, N J: Prentice Hall, 1995.
[23] Khasminskii R. Z. , Stochastic Stability of Differential Equations, Alphen aan den Rijn, The Netherlands: Sijthoff and Noorchoff, 1980.
[24] Mao X. , Exponential Stability of Stochastic Differential Equations, New York: Marcel Dekker Inc. , 1994.
[25] Arnold L. , Random Dynamical Systems, Berlin: Springer, 1998.
[26] 朱位秋,随机振动,北京:科学出版社,1992,1998.
[27] 庄表中等,非线性随机动理论及应用,浙江大学出版社,1986.
[28] 季文美、方同、陈松淇,机械振动,北京:科学出版社,1985.
[29] 欧进萍、王光远,结构随机振动,北京:高等教育出版社,1998.
[30] 方同,工程随机振动,北京:国防工业出版社,1995.
[31] 林家浩,张亚辉,随机振动的虚拟激励法,北京:科学出版社,2004.
[32] 朱位秋,非线性,随机动力学控制——Hamilton理论体系框架,北京:科学出版社,2003
[33] 朱位秋、黄志龙 应祖光,非线性随机动力学与控制哈密顿理论框架,力学与实践,2002,24 (3):1-9.
[35] 朱位秋,黄志龙,随机激励的耗散的Hamilton系统理论研究进展,力学与实践,2000,30:481-494
[36] W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamilton formulation, ASME, Appl. Mech. Rev. , in press.
[37] Bellomo N. and Casciati F, (eds), Nonlinear Stochastic Mechanics, Proc. IUTAM Symp. Berlin: Springer-Verlag, 1992.
[38] Naess A. and Krenk S. (eds), Advances in Nonlinear Stochastic Mechanics, Proc. IUTAM Symp. , Dordrecht: Kluwer Academic Publishers, 1996.
[39] Narayanan S. and lyengar R. N. , (eds), Nonlinearity and Stochastic Structueral Dynamics,
[2] Symposium on the Response of Nonliear Systems to Random Excitation, J. Acoust. Am. , 1963, 35: 1683—1721.
[3] Crandall, S. H. , and Mark, W. D. , Random Vibration in Mechanical Systems, Acadcmic Press, 1963.
[4] Robson, J. , D. , An Introduction to Random Vibration, Edinburgh University press, 1963. (中译本:J.D.罗伯逊,随机振动引论,湖南科学技术出版社,1980)
[5] Bolotin, V. V. , Statistical Methods in Stuctural Mechanics, Holden-Day, 1969.
[6] Newland, D. E, An Introduction to Random Vibration and Spectral Analysis, Longmans, 1stedit, 1975; 2nd edit, 1984. (第一版本中译本:D.E纽兰,随机振动与谱分析概论,机械工业版社,1980).
[7] Lin, Y. K. , Probabilistic Theory of Structural Dynamics, Mcgraw-Hill, 1967, (Repint by Robert E Krieger publishing Co, 1976).
[8] Nigam, N. C. , Introduction to Random Vibrations, The MIT press, 1983, (中译本:N.C.尼格姆,随机振动概论,上海交通大学出版社,1985).
[9] Crandall, S. H. , ed. , Random vibration, The MIT Press, 1963.
[10] Crandall S. H. and Zhu W. Q. Random Vibration: a survey of recent developments, ASME J. Appl. Mech. 50, 1983, 50th Anniversary Issue, 953-962.
[11] Spanos, P. D. , Stochastic linearization in Structural Dynamics, Appl. Mech. Rev. , 1981, 34: 1-8.
[12] Roberts, J. B. , and Spanos, R D. , Stochastic averaging: an approximate method of solving random vibration problems, Int. J, Non-Linear Mech. , 1986, 21: 111-134.
[13] Zhu, W. Q. , Stochassic averaging methods in random vibration, Appl. Mech. Rev. , 1988, 41 (5) : 189-199.
[14] Lin, Y. K. , and Cai, G. Q. , Vibration of nonlinear systems under additive and multiplicative random excitations, Appl. Mech. Rev. , 1989, 42 (11) : S135-S141.
[15] Bolotin V. V. , Random Vibration of Elastic Systems, Matinus Nijhoff Publishers, The Hague. 1984.
[16] Ibrahim R. A. , Parametric Random Vibration, Taunton: Reserch Studies Press LTD, 1985.
[17] Dimentberg M. F. , Statistical Dynamics of Nonlinear and Time-Varying Systems, Taunton: Reserch Sdudies Press LtD, England. , 1988.
[18] Roberts J. B. and Spanos P. D. , Random Vibration and Stctistical Linearization, New York: Wiley, 1990.
[19] Soong T. T. and Grigoriu M. , Random Vibration of Mechanical and Structural Systems, New Jersey: PTR Prentice-Hall, (1993),
[20] Soize C. , The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solution, Singapore: World Scientific, 1994.
[21] Lin Y. K. and Cai G. Q. , Probabilistic Structural Dynamics, Advanced Theory and Applications, New York: McGraw-Hill, 1995.
[22] Grigoriu M. , Applied Non-Gaussian Processes; Examples, Theory, Simulations, Linear Random Vibration and MATLAB Solutions, Englewood Cliffs, N J: Prentice Hall, 1995.
[23] Khasminskii R. Z. , Stochastic Stability of Differential Equations, Alphen aan den Rijn, The Netherlands: Sijthoff and Noorchoff, 1980.
[24] Mao X. , Exponential Stability of Stochastic Differential Equations, New York: Marcel Dekker Inc. , 1994.
[25] Arnold L. , Random Dynamical Systems, Berlin: Springer, 1998.
[26] 朱位秋,随机振动,北京:科学出版社,1992,1998.
[27] 庄表中等,非线性随机动理论及应用,浙江大学出版社,1986.
[28] 季文美、方同、陈松淇,机械振动,北京:科学出版社,1985.
[29] 欧进萍、王光远,结构随机振动,北京:高等教育出版社,1998.
[30] 方同,工程随机振动,北京:国防工业出版社,1995.
[31] 林家浩,张亚辉,随机振动的虚拟激励法,北京:科学出版社,2004.
[32] 朱位秋,非线性,随机动力学控制——Hamilton理论体系框架,北京:科学出版社,2003
[33] 朱位秋、黄志龙 应祖光,非线性随机动力学与控制哈密顿理论框架,力学与实践,2002,24 (3):1-9.
[35] 朱位秋,黄志龙,随机激励的耗散的Hamilton系统理论研究进展,力学与实践,2000,30:481-494
[36] W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamilton formulation, ASME, Appl. Mech. Rev. , in press.
[37] Bellomo N. and Casciati F, (eds), Nonlinear Stochastic Mechanics, Proc. IUTAM Symp. Berlin: Springer-Verlag, 1992.
[38] Naess A. and Krenk S. (eds), Advances in Nonlinear Stochastic Mechanics, Proc. IUTAM Symp. , Dordrecht: Kluwer Academic Publishers, 1996.
[39] Narayanan S. and lyengar R. N. , (eds), Nonlinearity and Stochastic Structueral Dynamics,