精细积分方法的改进及其在动力学与控制中的应用
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摘要
常微分方程组的数值计算一直是备受人们关注的领域,对此已发展了丰富的数值方法。近年来,精细积分方法得到广泛关注,已扩展到时变、非线性微分方程、偏微分方程的求解,并成功地应用到结构动力响应、随机振动、波导、热传导以及最优控制等领域,为不同领域的数值计算提供了一个高精度、高稳定性的算法平台,值得深入研究。另一方面,控制领域对数值计算的关注度和重要性意识正在加强,而合适的理论框架对于构造高性能算法有重要意义。现代控制论所奠基的状态空间法的起点至少也应回溯到Hamilton正则方程体系,表明经典力学与现代控制论有共同的数学形式和理论基础,两个学科的问题是相互对应的。因此,借鉴力学中成熟的有限元、子结构分析等方法,展开对最优控制领域数值方法和控制系统设计的研究是有意义的。本论文以发展高效、可靠的数值算法为主线,改进了精细积分算法平台的性能,研究了时滞、时变、非线性系统最优控制的数值计算和控制器设计等问题,开发了最优控制系统设计工具箱并将其应用于卫星编队飞行控制的研究。主要工作如下:
(1)采用矩阵函数逼近理论,提出了基于Pade级数逼近的矩阵指数精细积分方法中加权参数N和级数展开项数q的递推自适应选择算法,提高了精细积分方法的计算效率。并与MATLAB内置函数expm()进行了比较,表明本文方法在达到相同的效率的同时具有更高的精度和稳定性。
(2)提出了动力初值问题中非齐次项产生的Duhamel积分响应矩阵的扩展精细积分方法(EPIM),该方法不需对系统矩阵(或相关动力矩阵)求逆。当非齐次项为多项式函数、指数函数、正/余弦函数及其组合函数的形式时,可以得到计算机意义上的精确解。并推广应用于:1)与虚拟激励法结合,应用于随机振动响应的计算;2)结合传统数值积分技术(如Taylor级数单步法和Adams多步法),构造了求解非线性微分方程的显式/隐式算法;3)利用系数周期性变化的特点,导出了周期时变Floquet转移矩阵和一类非线性周期系统响应的计算格式;等。算例表明,基于扩展精细积分方法构造的算法提高了数值稳定性和适用范围,具有高效、高精度、高稳定性的优点。
(3)提出了两点边值问题中非齐次项产生的区段响应矩阵的扩展精细积分方法(EPIM),当非齐次项为多项式函数、指数函数、正/余弦函数及其组合函数的形式时,可以得到计算机意义上的精确解。在此基础上,研究了一般非齐次项的处理方法以及在无限长区段和变系数两点边值问题中的应用。还结合周期时变Floquet转移矩阵的扩展精细积分方法,导出了周期变系数Riccati、Lyapunov、Sylvester等矩阵微分方程的保结构算法,数值算例验证了算法的有效性。
(4)对时滞系统的H_∞最优控制和滤波进行了研究。首先采用扩展精细积分方法对连续时滞系统方程和性能指标离散化,以最大程度地保证与原系统的等价性。然后引入合适的增维向量,化为不显含时滞的标准离散形式,采用区段混合能方法和扩展W-W算法进行计算分析,增强了增维方法的可行性,从而为时滞H_∞最优控制和滤波系统的分析和设计提供了一套精确、稳定的算法。并导出了含输入时滞的H_∞全信息控制器,应用于建筑结构的减振控制,仿真显示对于不同的时滞量和地震激励形式,结构的振动响应都得到了有效抑制,验证了控制器的有效性。
(5)时变、非线性最优控制系统设计导出Hamilton系统两点边值问题,其数值算法应该保辛。本文在区段分析的框架下,提出了时变Hamilton两点边值问题基于常值精细积分的保辛摄动方法,导出了零阶、摄动系统分别基于区段混合能矩阵和区段传递矩阵的组合公式以及对应关系,指出前者具有内在的稳定性从而是更好的选择。进一步提出了时变非齐次Hamilton两点边值问题的保辛摄动方法,并应用于非线性最优控制问题的迭代计算,结果表明,迭代过程中关键算法的改进显著地提高了收敛速度,降低了对初始迭代值的敏感性,说明保辛摄动方法是一种高精度和稳定的算法。
(6)传统终端控制器往往存在终端高增益或奇异现象,只好在靠近终端区段采用开环控制。本文引入终端“软约束项”改进了性能指标,并利用Lagrange乘子的常数本质,构造了非奇异的、两个区段都具有反馈-前馈控制结构的终端控制器。分析了引入的“软约束项”对构造反馈结构控制器的重要影响,对于最小能量控制问题尤为重要。进一步利用区段混合能矩阵构造了反馈增益矩阵和控制系统方程的闭合解,导出了保结构递推算法,方便了控制器的设计与实现。并将该方法推广应用于离散时间系统的终端控制器设计。
(7)针对当前主流商业控制系统设计软件MATLAB缺乏有限长时间时变最优控制器设计功能的现状开发了PIMCSD Toolbox;在此基础上研究了典型双星编队重构的时变最优控制方案,研究成果为航天器编队控制系统的工程设计和应用提供了重要参考。
Numerical computation for a set of simultaneous ordinary differential equations(ODEs) is very important in applications. So far, tremendous efforts have been devoted to finding appropriate numerical methods to solve this problem. In recent years, precise integration method(PIM) for the numerical integration of ODEs has been proposed and has attracted a wide range of concerns. PIM not only can give a high accurate numerical result, which approaches the computer precision for the linear time-invariant ODEs, but also is free from stiff problems. It has been extended to solve time-variant, non-linear and partial differential equations and successfully applied to various fields, such as structural dynamics, random vibration, wave prorogation, transient heat conduction and optimal control, et al. The PIM provides a basic algorithm platform with high precision and high stability, so it is worthy of studying further. At the same time, numerical awareness in control needs to be increased. Especially, it is of great importance to choose an appropriate theoretical framework for constructing algorithms of high performance. The starting point of the state space method, i.e. the basis of modern control theory, should trace back at least to the Hamiltonian canonical equation system, which demonstrates that structural mechanics and optimal control have the same mathematical basis. Along this way of consideration, the mathematical problems of the two different fields have a one-to-one correspondence with each other. As a result, it is helpful to make further researches on the controller design and its numerical computation in optimal control field by introducing the mature methods in mechanics, such as finite element method, sub-structure techniques, et al. This dissertation aims at the development of efficient and reliable numerical methods, improves the performance of the PIM-based algorithm platform and studies the problems of the controller design and its numerical computation arising from the time-delay, time-varying, and non-linear optimal control systems. This dissertation also develops an optimal control system design and simulation toolbox (PEMCSD Toolbox) and applies it to the study of the formation flying control system. The main research work covers the following aspects:
(1) This dissertation presents an iterative algorithm for adaptive selection of the scaling parameter N and series-expanding parameter q by using the approximation theory of matrix functions, which plays an important role on the computational efficiency and numerical accuracy of the PIM based on Padéapproximation for matrix exponential. The proposed algorithm not only improves the computational efficiency, but also is independent of the matrix's characteristics. Numerical tests are made by comparing with the MATLAB's built-in function expm(), and it shows that the proposed algorithm achieves the same efficiency, but higher accuracy and stability.
(2) This dissertation presents the extended precise integration method(EPIM) for computing the response matrices of Duhamel integrations arising from non-homogenous dynamic systems. Numerical results of the response matrices can approach the computer precision when the non-homogenous terms are of the following forms, such as polynomial, exponential, trigonometric functions and their combinations. More importantly, the EPIM is independent of the quality of the system matrix(or its relative matrices) since it dose not need the inverse matrix calculation. The EPIM has been applied to some problems: 1) Combined with the pseudo-excitation method, an efficient and accurate algorithm for computing random responses of structural vibration has been proposed; 2) Combined with the traditional numerical integration techniques, such as the single-step method based on Taylor series expansion and the Adams multi-step method, several numerical algorithms with high efficiency and accuracy for the solution of non-linear differential equations have been constructed; 3)Making full use of the characteristics of periodicity, researches on computing the Floquet transition matrix of the periodic time-varying system and solving the responses of a class of periodic non-linear system have been made. Numerical examples show that the EPIM-based algorithms have advantages of high accuracy, high stability and simple formulas, which greatly improve the numerical stability and expand the scope of applications.
(3) The algebraic-equation method and interval-elimination method are derived based on the interval analysis techniques and the extended precise integration method(EPIM) for computing the interval response matrices, arising from the non-homogenous terms of the two point boundary value problems(TPBVPs), is proposed. The EPIM can give high precise numerical results approaching to computer precision for non-homogenous terms with certain special forms. High accurate and efficient algorithms based on the EPIM are constructed for the problems with general non-homogenous terms, infinite interval and time-varying coefficients. Finally structure-preserving algorithms for certain matrix differential equations with periodic coefficients, such as Riccati equations, Lyapunov equations and Sylvester equations, are constructed by combining interval analysis method with the EPIM for periodic Floquet transition matrix. Numerical results verify the effectiveness of the algorithms.
(4) Robust H_∞optimal control and filter of time-delay systems are studied in a uniform frame work. Firstly the continuous time-delay systems and its performance indices are discretized by the EPIM in order to ensure the equivalence to its original systems as much as possible. Then the discrete time-delay systems are transformed into standard discrete forms without time-delay by introducing the appropriate extended state vectors. So theories and methods of usual discrete system can be applied. Taking into account the increasing dimensions of the standardized system, the interval mixed energy method and extended W-W algorithms are introduced, which are with high parallelism and stability. So a set of accurate and stable algorithms are proposed for the computational problems of H_∞optimal control and filter systems. A H_∞full information controller with control time-delay is designed and applied to vibration attenuation of the seismic-excited buildings. Simulation results show that structural vibration is greatly attenuated for different amounts of time-delays and different types of seismic excitation, which verifies the effectiveness of the controller.
(5)This dissertation presents the symplectic-conservative perturbation method for computational problems of linear time-varying and non-linear optimal control systems. Since the necessary conditions of optimal control problems are equivalent to the TPBVPs of Hamiltonian systems, its numerical methods should be symplectic conservation. Firstly the symplectic conservative perturbation method based on the PIM is presented for the linear time-varying TPBVPs. Combination formulas between the zeros-order system and the perturbation system are derived based on the interval mixed energy matrices and the transition matrices, respectively. Their relationships are investigated and the former is found to be a better choice for optimal control problems because of its inherent stability. The symplectic-conservative perturbation method for non-homogenous time-varying Hamiltonian TPBVPs is further proposed and applied to the iterative computation of the non-linear optimal control problems. Numerical results show it not only increases convergency of the iteration algorithm but also decreases sensitivity to the initial iterative values greatly, which demonstrates that the proposed method is both high accurate and symplectic-conservative.
(6) Due to high feedback gains or singularity at the terminal time of traditional terminal controllers, an open-loop control for a short interval before the end time is often adopted. This dissertation presents a non-singular terminal controller with the feedback-feedforward architecture in both intervals, by introducing a new terminal "soft constraint" term to improve the variational formulas and using the essence of constant for the Lagrange multiplier. Influences of the "soft constraint" term on constructing the feedback controller are studied, which is of special importance for the minimal energy control. Closed-form solutions to the feedback matrices and states of the controlled system are constructed by introducing interval mixed energy matrices. Further more, structure-preserving algorithms are derived, which greatly facilitates the design and implementation of terminal controllers. The proposed method has been extended to the terminal controller of discrete-time systems successfully.
(7) The optimal control system design and simulation toolbox(PIMCSD Toolbox) is developed in view of the absence of functions on the finite time optimal control. Then time-varying controllers for the typical double satellites formation reconfiguration are studied based on the PIMCSD Toolbox. Research results provide an important reference for engineering designs and applications of spacecraft formation control systems.
(1)采用矩阵函数逼近理论,提出了基于Pade级数逼近的矩阵指数精细积分方法中加权参数N和级数展开项数q的递推自适应选择算法,提高了精细积分方法的计算效率。并与MATLAB内置函数expm()进行了比较,表明本文方法在达到相同的效率的同时具有更高的精度和稳定性。
(2)提出了动力初值问题中非齐次项产生的Duhamel积分响应矩阵的扩展精细积分方法(EPIM),该方法不需对系统矩阵(或相关动力矩阵)求逆。当非齐次项为多项式函数、指数函数、正/余弦函数及其组合函数的形式时,可以得到计算机意义上的精确解。并推广应用于:1)与虚拟激励法结合,应用于随机振动响应的计算;2)结合传统数值积分技术(如Taylor级数单步法和Adams多步法),构造了求解非线性微分方程的显式/隐式算法;3)利用系数周期性变化的特点,导出了周期时变Floquet转移矩阵和一类非线性周期系统响应的计算格式;等。算例表明,基于扩展精细积分方法构造的算法提高了数值稳定性和适用范围,具有高效、高精度、高稳定性的优点。
(3)提出了两点边值问题中非齐次项产生的区段响应矩阵的扩展精细积分方法(EPIM),当非齐次项为多项式函数、指数函数、正/余弦函数及其组合函数的形式时,可以得到计算机意义上的精确解。在此基础上,研究了一般非齐次项的处理方法以及在无限长区段和变系数两点边值问题中的应用。还结合周期时变Floquet转移矩阵的扩展精细积分方法,导出了周期变系数Riccati、Lyapunov、Sylvester等矩阵微分方程的保结构算法,数值算例验证了算法的有效性。
(4)对时滞系统的H_∞最优控制和滤波进行了研究。首先采用扩展精细积分方法对连续时滞系统方程和性能指标离散化,以最大程度地保证与原系统的等价性。然后引入合适的增维向量,化为不显含时滞的标准离散形式,采用区段混合能方法和扩展W-W算法进行计算分析,增强了增维方法的可行性,从而为时滞H_∞最优控制和滤波系统的分析和设计提供了一套精确、稳定的算法。并导出了含输入时滞的H_∞全信息控制器,应用于建筑结构的减振控制,仿真显示对于不同的时滞量和地震激励形式,结构的振动响应都得到了有效抑制,验证了控制器的有效性。
(5)时变、非线性最优控制系统设计导出Hamilton系统两点边值问题,其数值算法应该保辛。本文在区段分析的框架下,提出了时变Hamilton两点边值问题基于常值精细积分的保辛摄动方法,导出了零阶、摄动系统分别基于区段混合能矩阵和区段传递矩阵的组合公式以及对应关系,指出前者具有内在的稳定性从而是更好的选择。进一步提出了时变非齐次Hamilton两点边值问题的保辛摄动方法,并应用于非线性最优控制问题的迭代计算,结果表明,迭代过程中关键算法的改进显著地提高了收敛速度,降低了对初始迭代值的敏感性,说明保辛摄动方法是一种高精度和稳定的算法。
(6)传统终端控制器往往存在终端高增益或奇异现象,只好在靠近终端区段采用开环控制。本文引入终端“软约束项”改进了性能指标,并利用Lagrange乘子的常数本质,构造了非奇异的、两个区段都具有反馈-前馈控制结构的终端控制器。分析了引入的“软约束项”对构造反馈结构控制器的重要影响,对于最小能量控制问题尤为重要。进一步利用区段混合能矩阵构造了反馈增益矩阵和控制系统方程的闭合解,导出了保结构递推算法,方便了控制器的设计与实现。并将该方法推广应用于离散时间系统的终端控制器设计。
(7)针对当前主流商业控制系统设计软件MATLAB缺乏有限长时间时变最优控制器设计功能的现状开发了PIMCSD Toolbox;在此基础上研究了典型双星编队重构的时变最优控制方案,研究成果为航天器编队控制系统的工程设计和应用提供了重要参考。
Numerical computation for a set of simultaneous ordinary differential equations(ODEs) is very important in applications. So far, tremendous efforts have been devoted to finding appropriate numerical methods to solve this problem. In recent years, precise integration method(PIM) for the numerical integration of ODEs has been proposed and has attracted a wide range of concerns. PIM not only can give a high accurate numerical result, which approaches the computer precision for the linear time-invariant ODEs, but also is free from stiff problems. It has been extended to solve time-variant, non-linear and partial differential equations and successfully applied to various fields, such as structural dynamics, random vibration, wave prorogation, transient heat conduction and optimal control, et al. The PIM provides a basic algorithm platform with high precision and high stability, so it is worthy of studying further. At the same time, numerical awareness in control needs to be increased. Especially, it is of great importance to choose an appropriate theoretical framework for constructing algorithms of high performance. The starting point of the state space method, i.e. the basis of modern control theory, should trace back at least to the Hamiltonian canonical equation system, which demonstrates that structural mechanics and optimal control have the same mathematical basis. Along this way of consideration, the mathematical problems of the two different fields have a one-to-one correspondence with each other. As a result, it is helpful to make further researches on the controller design and its numerical computation in optimal control field by introducing the mature methods in mechanics, such as finite element method, sub-structure techniques, et al. This dissertation aims at the development of efficient and reliable numerical methods, improves the performance of the PIM-based algorithm platform and studies the problems of the controller design and its numerical computation arising from the time-delay, time-varying, and non-linear optimal control systems. This dissertation also develops an optimal control system design and simulation toolbox (PEMCSD Toolbox) and applies it to the study of the formation flying control system. The main research work covers the following aspects:
(1) This dissertation presents an iterative algorithm for adaptive selection of the scaling parameter N and series-expanding parameter q by using the approximation theory of matrix functions, which plays an important role on the computational efficiency and numerical accuracy of the PIM based on Padéapproximation for matrix exponential. The proposed algorithm not only improves the computational efficiency, but also is independent of the matrix's characteristics. Numerical tests are made by comparing with the MATLAB's built-in function expm(), and it shows that the proposed algorithm achieves the same efficiency, but higher accuracy and stability.
(2) This dissertation presents the extended precise integration method(EPIM) for computing the response matrices of Duhamel integrations arising from non-homogenous dynamic systems. Numerical results of the response matrices can approach the computer precision when the non-homogenous terms are of the following forms, such as polynomial, exponential, trigonometric functions and their combinations. More importantly, the EPIM is independent of the quality of the system matrix(or its relative matrices) since it dose not need the inverse matrix calculation. The EPIM has been applied to some problems: 1) Combined with the pseudo-excitation method, an efficient and accurate algorithm for computing random responses of structural vibration has been proposed; 2) Combined with the traditional numerical integration techniques, such as the single-step method based on Taylor series expansion and the Adams multi-step method, several numerical algorithms with high efficiency and accuracy for the solution of non-linear differential equations have been constructed; 3)Making full use of the characteristics of periodicity, researches on computing the Floquet transition matrix of the periodic time-varying system and solving the responses of a class of periodic non-linear system have been made. Numerical examples show that the EPIM-based algorithms have advantages of high accuracy, high stability and simple formulas, which greatly improve the numerical stability and expand the scope of applications.
(3) The algebraic-equation method and interval-elimination method are derived based on the interval analysis techniques and the extended precise integration method(EPIM) for computing the interval response matrices, arising from the non-homogenous terms of the two point boundary value problems(TPBVPs), is proposed. The EPIM can give high precise numerical results approaching to computer precision for non-homogenous terms with certain special forms. High accurate and efficient algorithms based on the EPIM are constructed for the problems with general non-homogenous terms, infinite interval and time-varying coefficients. Finally structure-preserving algorithms for certain matrix differential equations with periodic coefficients, such as Riccati equations, Lyapunov equations and Sylvester equations, are constructed by combining interval analysis method with the EPIM for periodic Floquet transition matrix. Numerical results verify the effectiveness of the algorithms.
(4) Robust H_∞optimal control and filter of time-delay systems are studied in a uniform frame work. Firstly the continuous time-delay systems and its performance indices are discretized by the EPIM in order to ensure the equivalence to its original systems as much as possible. Then the discrete time-delay systems are transformed into standard discrete forms without time-delay by introducing the appropriate extended state vectors. So theories and methods of usual discrete system can be applied. Taking into account the increasing dimensions of the standardized system, the interval mixed energy method and extended W-W algorithms are introduced, which are with high parallelism and stability. So a set of accurate and stable algorithms are proposed for the computational problems of H_∞optimal control and filter systems. A H_∞full information controller with control time-delay is designed and applied to vibration attenuation of the seismic-excited buildings. Simulation results show that structural vibration is greatly attenuated for different amounts of time-delays and different types of seismic excitation, which verifies the effectiveness of the controller.
(5)This dissertation presents the symplectic-conservative perturbation method for computational problems of linear time-varying and non-linear optimal control systems. Since the necessary conditions of optimal control problems are equivalent to the TPBVPs of Hamiltonian systems, its numerical methods should be symplectic conservation. Firstly the symplectic conservative perturbation method based on the PIM is presented for the linear time-varying TPBVPs. Combination formulas between the zeros-order system and the perturbation system are derived based on the interval mixed energy matrices and the transition matrices, respectively. Their relationships are investigated and the former is found to be a better choice for optimal control problems because of its inherent stability. The symplectic-conservative perturbation method for non-homogenous time-varying Hamiltonian TPBVPs is further proposed and applied to the iterative computation of the non-linear optimal control problems. Numerical results show it not only increases convergency of the iteration algorithm but also decreases sensitivity to the initial iterative values greatly, which demonstrates that the proposed method is both high accurate and symplectic-conservative.
(6) Due to high feedback gains or singularity at the terminal time of traditional terminal controllers, an open-loop control for a short interval before the end time is often adopted. This dissertation presents a non-singular terminal controller with the feedback-feedforward architecture in both intervals, by introducing a new terminal "soft constraint" term to improve the variational formulas and using the essence of constant for the Lagrange multiplier. Influences of the "soft constraint" term on constructing the feedback controller are studied, which is of special importance for the minimal energy control. Closed-form solutions to the feedback matrices and states of the controlled system are constructed by introducing interval mixed energy matrices. Further more, structure-preserving algorithms are derived, which greatly facilitates the design and implementation of terminal controllers. The proposed method has been extended to the terminal controller of discrete-time systems successfully.
(7) The optimal control system design and simulation toolbox(PIMCSD Toolbox) is developed in view of the absence of functions on the finite time optimal control. Then time-varying controllers for the typical double satellites formation reconfiguration are studied based on the PIMCSD Toolbox. Research results provide an important reference for engineering designs and applications of spacecraft formation control systems.
引文
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