受随机激励工程结构振动控制方法研究
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摘要
随机性是自然界最基本的规律之一,大量军事或民用设施都处于地震、风、海浪、流冰、不平路面激励等随机环境中,从而发生危及设施与人员安全性、舒适性、设施的正常运行或疲劳寿命等十分有害的随机振动。如何在设计阶段就努力将结构的随机振动控制在可以接受的限度以内,是工程界长期以来努力追求的目标。可是迄今为止所取得的成就还远不能满足工程界的需求。在本论文的研究中,密切结合对随机振动控制的需求比较突出的建筑抗震、汽车振动等工程问题,不但融入了近十几年来在数学、力学和控制理论界所出现的一些创新成果,并且采取各种新的思路对它们进行了有针对性的发展和再创新,提出了一系列实施结构随机振动控制的有效方法。这些方法和通过研究并使用这些方法而得到的结论,对于其它工程领域的随机振动控制也是有实用或借鉴意义的。
建筑结构所受的地震作用和车辆行驰时所受的路面激励都是随机荷载。由它们所造成的结构振动本应按随机振动方法来处理。但由于常规随机振动方法的复杂低效,以往在研究建筑结构抗震控制器或车辆主动悬架控制器时,数值仿真一般都只在效率比较低的时域内进行直接数值积分。如果要对控制前后系统响应的统计特性作较为精确的估计,往往需要采用很多条地震加速度记录或路面不平度样本,并取很小的时间步长进行数值积分,从而耗费大量的计算时间,同时不得不采用十分简化的结构模型,使得控制效果大受影响。本论文在处理这类控制问题时,直接在随机振动方法框架内进行。尤其是将近年来由我国学者提出的虚拟激励法、精细积分法和在哈密顿体系内保辛、保性能,和保持高鲁棒性等措施引入随机振动控制领域,由地震激励功率谱密度或路面功率谱密度直接计算出系统各种平稳或非平稳随机响应的功率谱密度,既精确可靠,又快速方便,并且控制前后建筑结构的地震响应或车辆乘员的舒适度等指标都很容易得到。因此,本论文直接基于平稳/非平稳随机振动虚拟激励、精细积分等理论和方法对控制器进行研究是颇具特色的,具有很强的探索性和应用前景。
本论文在研究新的建筑结构抗震控制器和车辆主动悬架控制器设计方法时,还尽可能地基于线性矩阵不等式处理框架,由此充分利用现有的凸优化技术而使控制器的设计简便易行。本论文根据建筑结构抗震控制和车辆主动悬架设计的特点,基于线性矩阵不等式处理方法,提出了新的建筑结构抗震鲁棒H_∞控制策略、保性能控制策略和车辆主动悬架鲁棒H_2/H_∞控制策略、考虑时域硬约束的H_∞控制策略。
具体说来,本论文主要进行了以下几方面的研究工作:
(1)扩展了LQG控制方法的求解途径和应用范围。由于常规的非平稳随机振动算法非常复杂低效,在结构抗震LQG控制问题中,地震的非平稳随机性以往极少被考虑。本文以毗邻建筑避碰控制为例,对非平稳随机地震激励下的LQG控制问题的求解进行了新的尝试。通过引入区段混合能矩阵,对微分Riccati方程进行了精细求解,并借助虚拟激励法及在状态空间中的时域精细积分法,使这类以前难于处理的时变LQG控制问题得到了高精度、高效率的解决。
(2)基于线性矩阵不等式处理方法,提出了新的鲁棒风控制方法、给定二次型性能指标下的保性能控制方法和鲁棒H_2/H_∞控制方法,并应用于建筑结构抗震和车辆主动悬架设计。在控制器设计过程中,忽略结构模型参数的不确定性可能会导致控制系统性能恶化甚至失稳。本文基于线性矩阵不等式处理方法,提出并证明了考虑参数不确定性的鲁棒H_∞输出反馈控制器、保性能输出反馈控制器和鲁棒H_2/H_∞输出反馈控制器存在的充分条件。大量数值仿真结果表明,当建筑结构的刚度、阻尼、质量或车辆的悬挂质量存在变异时,本文给出的这三种控制策略都能实现很好的控制效果。
(3)基于线性矩阵不等式处理方法,提出了一种新的考虑时域硬约束条件的H_∞输出反馈控制器设计方法,并应用于车辆主动悬架设计。车辆的操纵稳定性和乘坐舒适性对悬架各项性能有着不同的要求,为了平衡相互矛盾的诸项性能要求,本文从提高车辆乘坐舒适性入手,以车身加速度响应为控制输出向量,以悬架动行程、轮胎动载荷和作动器输出控制力响应为约束输出向量,分别基于连续时间系统和离散时间系统模型提出并证明了时域硬约束条件下H_∞输出反馈控制器存在的充分条件。
(4)基于精细积分方法和保辛摄动思想,提出了一种新的时变系统LQG控制精细算法,并用于车桥耦合系统的垂向减振控制。时变系统LQG控制器设计中,需要求解变系数矩阵Riccati微分方程和变系数矩阵Kalman-Bucy滤波方程。通过将原时变Hamilton系统分解为零阶近似和保辛摄动两个Hamilton系统,引入区段混合能,给出了Riccati方程和Kalman-Bucy滤波方程状态转移矩阵基于区段矩阵合并的递推求解格式。以控制车辆过桥时的平顺性为例进行数值仿真,证实了此方法的优越性。
Randomness is one of the most fundamental laws in nature. Many militarian equipment and civil engineering structures are subjected to random environmental loads, such as earthquakes, wind gusts, ocean waves, drift ices or road irregularities. Random vibration induced by these environmental excitations may lead to structural fatigue damage, abnormal facility operation, uncomfortable ride, or safety problems for workers. How to control such harmful random vibrations within acceptable levels has long been a common target for engineering communities. Unfortunately, achievements so far obtained in this field are still quite limited, far beyond the requirements of engineering societies. In this dissertation, aiming at some urgent requirements in building earthquake-resistance and vehicle vibration, the author uses some recent achievements in mathematics, mechanics and control theory, combine and further develop them to generate a series of innovative and effective approaches in order to deal with such random vibration based structural control problems. These approaches and the conclusions drawn by using them in the study of the above civil and vehicle engineering problems have been fully justified in this dissertation. And they are also useful for the random vibration control in other engineering fields.
Earthquakes and road irregularities are random and they can reasonably be treated as stochastic processes in the controller designs. However, as the usual random vibration methods are too complex and lack of efficiency, in the past time-domain based direct integration methods were usually adopted in the simulations, with very small time step and simplified structure model.
In it a substantial number of earthquake acceleration records or excitation samples of the road inputs must be used in order to get statistical characteristics of the system's responses, which greatly increases the required computational effort. In this dissertation, some symplectic conservative, guaranteed cost and high robust methods are introduced in the field of random vibration control. Random responses of the uncontrolled and controlled structures are calculated with pseudo excitation method (PEM) and precise integration method (PIM) used. PEM is a highly efficient and accurate probabilistic method, in it the power spectral densities (PSDs) of the structural responses are calculated directly and conveniently from the ground acceleration or road surface elevation PSD. This yields accurate root mean squares of the structural responses, including the frequency-weighted acceleration root mean squares of the vehicle body. In a word, one feature of this dissertation is the application of stationary/non-stationary random vibration theory to the structural vibration control with PEM and PID used.
Linear matrix inequality (LMI) approach has emerged as a powerful formulation and design technique for a variety of linear control problems, for which the existing convex optimization techniques such as interior-point algorithms, can be used effectively and conveniently. For new control strategies proposed in this dissertation, LMI optimization approach has been widely used. These strategies, including robust H_∞control, guaranteed cost control, robust H_2/H_∞control and constrained H_∞, control, are applied to building's aseismic control or active suspension design.
Main research work of this dissertation can be summarized as follows:
(1) Field of application of LQG method and its solutions are expanded. LQG regulators have been used in many engineering fields, however, as the low efficiency of usual random vibration analysis methods, LQG control technique still requires further development especially when the disturbances are non-stationary. In this dissertation, a new attempt is studied in which the LQG control is applied to adjacent tall buildings subjected to non-stationary seismic random excitations. With interval mixed variable energy introduced, the Riccati differential equation is solved precisely via the combination of interval matrices. With PEM and PIM used, such difficult transient LQG control process is solved with high precision and efficiency.
(2) Based on LMIs, a new robust H_∞control approach, a new guaranteed cost control approach and a new robust H_2/H_∞control approach are presented, which are applied to aseismic structures or active suspension with uncertainties in model parameters. Uncertainties in the modeling of structures always exist. Neglecting these uncertainties may cause degradation and even instability of controlled structural systems. In this dissertation, based on LMIs, sufficient conditions for the existence of such output feedback controllers are derived, while the specific steps for controller designing are suggested. Numerical results show that whether the variation of the structural stiffness, damping, mass parameters or sprung mass of the vehicles exists or not, the proposed robust controllers behave very satisfactorily.
(3) Based on LMIs, a constrained H_∞output feedback control method for active suspensions is proposed. Usually, requirements for advanced vehicle suspensions are conflicting in order to improve driving safety and ride comfort. This dissertation formulates the active suspension control problem as a constrained H_∞, output feedback control problem by choosing body accelerations as the controlled output and specifying the suspension stroke, dynamic tire load and control force as time-domain hard constraints. Such constrained H_∞, output feedback controllers are investigated for continuous-time system and discrete-time system respectively. Sufficient conditions for the existence of the controllers, as well as the detail steps for the controller designing, are given.
(4) Based on the precise integration method and symplectic conservative perturbation method, a new precise algorithm is proposed for linear quadratic Gaussian (LQG) control of time-varying systems, which is applied to the vertical vibration suppressing of a rail carriage moving on a simply supported girder bridge. Vehicle-bridge coupling systems are time-dependent, which lead to the time-varying Riccati differential equation and the time-varying Kalman-Bucy filter equation, both of them need be solved in the LQG controller design. In this dissertation, the original time-varying Hamiltonian system, which corresponds to the time-varying Riccati differential equation, is decomposed into two Hamiltonian systems, i.e. a zero-order system and a residual perturbation system via the canonical transformation. With interval mixed variable energy introduced, the Riccati equations are solved recursively via combinations of interval matrices, as well as the state transfer matrices of the Kalman-Bucy filter equation. Its superiority has also been justified via numerical simulations of the random vibration control of a vehicle-bridge coupling system caused by irregular bridge surfaces.
建筑结构所受的地震作用和车辆行驰时所受的路面激励都是随机荷载。由它们所造成的结构振动本应按随机振动方法来处理。但由于常规随机振动方法的复杂低效,以往在研究建筑结构抗震控制器或车辆主动悬架控制器时,数值仿真一般都只在效率比较低的时域内进行直接数值积分。如果要对控制前后系统响应的统计特性作较为精确的估计,往往需要采用很多条地震加速度记录或路面不平度样本,并取很小的时间步长进行数值积分,从而耗费大量的计算时间,同时不得不采用十分简化的结构模型,使得控制效果大受影响。本论文在处理这类控制问题时,直接在随机振动方法框架内进行。尤其是将近年来由我国学者提出的虚拟激励法、精细积分法和在哈密顿体系内保辛、保性能,和保持高鲁棒性等措施引入随机振动控制领域,由地震激励功率谱密度或路面功率谱密度直接计算出系统各种平稳或非平稳随机响应的功率谱密度,既精确可靠,又快速方便,并且控制前后建筑结构的地震响应或车辆乘员的舒适度等指标都很容易得到。因此,本论文直接基于平稳/非平稳随机振动虚拟激励、精细积分等理论和方法对控制器进行研究是颇具特色的,具有很强的探索性和应用前景。
本论文在研究新的建筑结构抗震控制器和车辆主动悬架控制器设计方法时,还尽可能地基于线性矩阵不等式处理框架,由此充分利用现有的凸优化技术而使控制器的设计简便易行。本论文根据建筑结构抗震控制和车辆主动悬架设计的特点,基于线性矩阵不等式处理方法,提出了新的建筑结构抗震鲁棒H_∞控制策略、保性能控制策略和车辆主动悬架鲁棒H_2/H_∞控制策略、考虑时域硬约束的H_∞控制策略。
具体说来,本论文主要进行了以下几方面的研究工作:
(1)扩展了LQG控制方法的求解途径和应用范围。由于常规的非平稳随机振动算法非常复杂低效,在结构抗震LQG控制问题中,地震的非平稳随机性以往极少被考虑。本文以毗邻建筑避碰控制为例,对非平稳随机地震激励下的LQG控制问题的求解进行了新的尝试。通过引入区段混合能矩阵,对微分Riccati方程进行了精细求解,并借助虚拟激励法及在状态空间中的时域精细积分法,使这类以前难于处理的时变LQG控制问题得到了高精度、高效率的解决。
(2)基于线性矩阵不等式处理方法,提出了新的鲁棒风控制方法、给定二次型性能指标下的保性能控制方法和鲁棒H_2/H_∞控制方法,并应用于建筑结构抗震和车辆主动悬架设计。在控制器设计过程中,忽略结构模型参数的不确定性可能会导致控制系统性能恶化甚至失稳。本文基于线性矩阵不等式处理方法,提出并证明了考虑参数不确定性的鲁棒H_∞输出反馈控制器、保性能输出反馈控制器和鲁棒H_2/H_∞输出反馈控制器存在的充分条件。大量数值仿真结果表明,当建筑结构的刚度、阻尼、质量或车辆的悬挂质量存在变异时,本文给出的这三种控制策略都能实现很好的控制效果。
(3)基于线性矩阵不等式处理方法,提出了一种新的考虑时域硬约束条件的H_∞输出反馈控制器设计方法,并应用于车辆主动悬架设计。车辆的操纵稳定性和乘坐舒适性对悬架各项性能有着不同的要求,为了平衡相互矛盾的诸项性能要求,本文从提高车辆乘坐舒适性入手,以车身加速度响应为控制输出向量,以悬架动行程、轮胎动载荷和作动器输出控制力响应为约束输出向量,分别基于连续时间系统和离散时间系统模型提出并证明了时域硬约束条件下H_∞输出反馈控制器存在的充分条件。
(4)基于精细积分方法和保辛摄动思想,提出了一种新的时变系统LQG控制精细算法,并用于车桥耦合系统的垂向减振控制。时变系统LQG控制器设计中,需要求解变系数矩阵Riccati微分方程和变系数矩阵Kalman-Bucy滤波方程。通过将原时变Hamilton系统分解为零阶近似和保辛摄动两个Hamilton系统,引入区段混合能,给出了Riccati方程和Kalman-Bucy滤波方程状态转移矩阵基于区段矩阵合并的递推求解格式。以控制车辆过桥时的平顺性为例进行数值仿真,证实了此方法的优越性。
Randomness is one of the most fundamental laws in nature. Many militarian equipment and civil engineering structures are subjected to random environmental loads, such as earthquakes, wind gusts, ocean waves, drift ices or road irregularities. Random vibration induced by these environmental excitations may lead to structural fatigue damage, abnormal facility operation, uncomfortable ride, or safety problems for workers. How to control such harmful random vibrations within acceptable levels has long been a common target for engineering communities. Unfortunately, achievements so far obtained in this field are still quite limited, far beyond the requirements of engineering societies. In this dissertation, aiming at some urgent requirements in building earthquake-resistance and vehicle vibration, the author uses some recent achievements in mathematics, mechanics and control theory, combine and further develop them to generate a series of innovative and effective approaches in order to deal with such random vibration based structural control problems. These approaches and the conclusions drawn by using them in the study of the above civil and vehicle engineering problems have been fully justified in this dissertation. And they are also useful for the random vibration control in other engineering fields.
Earthquakes and road irregularities are random and they can reasonably be treated as stochastic processes in the controller designs. However, as the usual random vibration methods are too complex and lack of efficiency, in the past time-domain based direct integration methods were usually adopted in the simulations, with very small time step and simplified structure model.
In it a substantial number of earthquake acceleration records or excitation samples of the road inputs must be used in order to get statistical characteristics of the system's responses, which greatly increases the required computational effort. In this dissertation, some symplectic conservative, guaranteed cost and high robust methods are introduced in the field of random vibration control. Random responses of the uncontrolled and controlled structures are calculated with pseudo excitation method (PEM) and precise integration method (PIM) used. PEM is a highly efficient and accurate probabilistic method, in it the power spectral densities (PSDs) of the structural responses are calculated directly and conveniently from the ground acceleration or road surface elevation PSD. This yields accurate root mean squares of the structural responses, including the frequency-weighted acceleration root mean squares of the vehicle body. In a word, one feature of this dissertation is the application of stationary/non-stationary random vibration theory to the structural vibration control with PEM and PID used.
Linear matrix inequality (LMI) approach has emerged as a powerful formulation and design technique for a variety of linear control problems, for which the existing convex optimization techniques such as interior-point algorithms, can be used effectively and conveniently. For new control strategies proposed in this dissertation, LMI optimization approach has been widely used. These strategies, including robust H_∞control, guaranteed cost control, robust H_2/H_∞control and constrained H_∞, control, are applied to building's aseismic control or active suspension design.
Main research work of this dissertation can be summarized as follows:
(1) Field of application of LQG method and its solutions are expanded. LQG regulators have been used in many engineering fields, however, as the low efficiency of usual random vibration analysis methods, LQG control technique still requires further development especially when the disturbances are non-stationary. In this dissertation, a new attempt is studied in which the LQG control is applied to adjacent tall buildings subjected to non-stationary seismic random excitations. With interval mixed variable energy introduced, the Riccati differential equation is solved precisely via the combination of interval matrices. With PEM and PIM used, such difficult transient LQG control process is solved with high precision and efficiency.
(2) Based on LMIs, a new robust H_∞control approach, a new guaranteed cost control approach and a new robust H_2/H_∞control approach are presented, which are applied to aseismic structures or active suspension with uncertainties in model parameters. Uncertainties in the modeling of structures always exist. Neglecting these uncertainties may cause degradation and even instability of controlled structural systems. In this dissertation, based on LMIs, sufficient conditions for the existence of such output feedback controllers are derived, while the specific steps for controller designing are suggested. Numerical results show that whether the variation of the structural stiffness, damping, mass parameters or sprung mass of the vehicles exists or not, the proposed robust controllers behave very satisfactorily.
(3) Based on LMIs, a constrained H_∞output feedback control method for active suspensions is proposed. Usually, requirements for advanced vehicle suspensions are conflicting in order to improve driving safety and ride comfort. This dissertation formulates the active suspension control problem as a constrained H_∞, output feedback control problem by choosing body accelerations as the controlled output and specifying the suspension stroke, dynamic tire load and control force as time-domain hard constraints. Such constrained H_∞, output feedback controllers are investigated for continuous-time system and discrete-time system respectively. Sufficient conditions for the existence of the controllers, as well as the detail steps for the controller designing, are given.
(4) Based on the precise integration method and symplectic conservative perturbation method, a new precise algorithm is proposed for linear quadratic Gaussian (LQG) control of time-varying systems, which is applied to the vertical vibration suppressing of a rail carriage moving on a simply supported girder bridge. Vehicle-bridge coupling systems are time-dependent, which lead to the time-varying Riccati differential equation and the time-varying Kalman-Bucy filter equation, both of them need be solved in the LQG controller design. In this dissertation, the original time-varying Hamiltonian system, which corresponds to the time-varying Riccati differential equation, is decomposed into two Hamiltonian systems, i.e. a zero-order system and a residual perturbation system via the canonical transformation. With interval mixed variable energy introduced, the Riccati equations are solved recursively via combinations of interval matrices, as well as the state transfer matrices of the Kalman-Bucy filter equation. Its superiority has also been justified via numerical simulations of the random vibration control of a vehicle-bridge coupling system caused by irregular bridge surfaces.
引文
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