结构时域辨识方法及传感器优化布置问题研究
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摘要
对重要土木工程结构进行健康监测和状态评估,是当前世界范围内的热点课题；而包括参数识别与荷载识别两类逆问题在内的结构动力学系统辨识技术,是结构健康监测与状态评估理论的核心内容。近几十年来,国内外学者在这一领域开展了大量研究工作,提出了许多理论与算法,主要分为频域法、时域法以及在此基础上发展出来的其他方法。与频域法相比,时域法直接利用时域信号进行辨识,在工程实际应用中更为方便,所以近年来得到广泛关注,取得大量研究成果。然而,受结构复杂性及环境干扰等因素的影响和制约,这些成果在实际应用中还存在一些有待解决的问题,如输出数据的不完备性、测量噪声和模型误差等的不确定性以及逆问题的不适定性等,都会对辨识精度产生不利影响。此外,系统辨识之前需要对结构进行动力测试,但测试传感器只能布设在有限结构位置上,传感器布置的合理与否会对辨识结果产生重要影响。
针对上述问题,本文开展了时域系统辨识问题的算法优化及传感器优化布置方法的研究。论文的主要工作和取得的成果如下：
(1)基于时域动态荷载识别方程的不适定分析,从识别方程的性态出发,提出了一种新的传感器优化布置准则——最小不适定性准则,并基于该准则提出了两种传感器数目确定条件下的位置优化方法：一种是基于结构系统马尔科夫参数矩阵条件数的直接算法,其缺点是当可能的传感器组合数目较大时,计算较为耗时：另一种是基于马尔科夫参数矩阵相关性分析的快速算法,定义了可以描述马尔科夫参数矩阵性态的相关性矩阵及传感器布置的优化指标。数值模拟结果表明,由两种传感器优化布置方法确定的最优传感器布置均可获得稳定性好、计算精度高的荷载识别结果,可用于解决时域动态荷载识别的传感器优化布置问题；随着备选传感器组合数目的增加,直接算法的计算时长会显著增加,而快速算法几乎不变,计算效率明显占优。
(2)基于转换矩阵的概念,将动态荷载识别的状态空间法拓展成了外界激励未知条件下的结构时域响应重构方法,仅利用部分测点的动态响应,通过转换矩阵重构出其他未测试位置处的响应,可用于解决时域辨识中输出数据不完备的问题。此外,还提出了一种传感器两步布设法：第一步,以重构方程具有稳定解为目标,基于单边Jacobi变换法和QR正交三角分解对全部备选测点对应的马尔科夫参数矩阵进行奇异值分解,将非零奇异值对应的传感器位置作为初始传感器布置；第二步,采用逐步积累法,以噪声效应放大指标最小为目标,在初始布置的基础上逐步增加传感器,直至达到收敛要求后获得最终传感器布置。数值模拟结果表明,该方法可根据工程实际需要,在保证重构方程具有稳定解的前提下,灵活确定最终传感器布置,获得所需的重构精度。
(3)针对振动响应灵敏度损伤识别方法,提出了一种修正Tikhonov正则化方法,可用于解决同时考虑测量噪声和模型误差干扰的条件下,传统Tikhonov正则化解不易收敛的问题。首先,对边界约束实施阈值控制,以保证解的物理意义;其次,对确定正则化参数的L-曲线方法进行修正；再次,对测量响应进行切比雪夫多项式去噪处理,减小噪声对识别结果的不利影响。数值模拟结果表明,当同时考虑噪声干扰和模型误差时,修正Tikhonov正则化方法可以使待识别的结构刚度参数逐渐收敛到一个相对正确的路径上,其损伤识别精度明显优于传统正则化方法。
(4)针对振动响应灵敏度损伤识别方法,提出了一种基于多重优化目标的传感器优化布置方法。首先,推导了结构刚度差异参数对三种典型不确定性因素——模型误差、测量噪声和荷载误差的灵敏度,进而得到不同因素所对应的识别误差协方差矩阵；然后,基于识别误差最小准则,定义了考虑多重不确定性因素的目标函数,并采用启发式搜索算法,获得了多重优化目标问题的Pareto最优解。数值模拟结果表明,考虑多重不确定性因素的条件下,由该方法确定的最优传感器布置,其损伤识别的准确性和可靠性均比较高。
Structural health monitoring and condition assessment on important civil engineering structures is an active area of research. Structural system identification including parameter identification and force identification plays an important role in structural health monitoring and condition assessment. Much attention has been devoted to this area over the last decades and various methods have been proposed, which are either in the frequency domain or time domain. Increasing interests have been focused on time domain methods in recent years, because comparing to frequency domain methods, time domain methods can directly use the measured signal and they are much easier for practical application. However, duo to the complexity of the structure and influences of environmental perturbations, there are still some problems needed to be solved, e.g. incomplete measurement data, multiple uncertainties such as noise and model errors and ill-conditioning of the identification equation etc., have adverse effects on identification accuracy. Moreover, dynamic tests should be carried out before structural system identification, and the degrees of freedom (DOFs) with sensors are limited comparing to the DOFs of the whole structure. Thus, the accuracy of system identification may vary significantly with different spatial location of the response measurements.
The methods of algorithm optimization and optimal sensor placement for structural system identification including force identification and damage identification in time domain are presented in this dissertation. The main contents and achievements are as follows:
(1) A new criterion for optimal sensor placement is presented based on the ill-conditioning analysis of the force identification equation in state space, and it is called criterion of the minimization of ill-conditioning. Two different sensor placement methods with determined number of sensor based on the proposed criterion are presented. The first one is based on direct computation of the condition number of the system Markov parameter matrix, and it would be time consuming when a large number of candidate combinations of sensor locations is considered. The second approach is based on the correlation analysis of the system Markov parameter matrix. A sensor correlation matrix is defined and the correlation criterion, which can indicate the ill-conditioning of the Markov parameter matrix, is introduced. Results from numerical simulations reveal that the performances of both methods are similar when the number of candidate combination of sensors is small. However, when there are a large number of candidate combinations, the method based on correlation analysis of the Markov parameter matrix performs better with consistently good sensor placement for force identification and much less computation effort.
(2) Dynamic force identification in state space is transformed to structural dynamic response reconstruction. The unmeasured structural responses can be reconstructed from limited measured responses. A new two-step sensor placement method is proposed for better prediction of the dynamic response reconstruction. In the first step, the system Markov parameter matrix corresponding to candidate sensor locations are singular value decomposed with One-sided Jacobi-transformation and QR decomposition methods. Sensor locations with non-zero singular values are combined as the initial sensor combination. In the second step, a measurement noise effect index is defined and the number and locations for the final sensor placement can be obtained from a heuristic forward sequential sensor placement algorithm based on the minimization of the noise effect index. Results of numerical simulations reveal that the sensors selected from the proposed method would lead to acceptable error of response reconstruction even with measurement noise.
(3) Damage identification equation based on sensitivity approach from the dynamic responses is ill-conditioned and is usually solved with regularization method. When the structural system contains measurement noise and model errors, the identification results from Tikhonov Regularization method often diverge after several iterations. A Modified Tikhonov Regularization method is presented to solve the above problem. New side conditions with limits on the identification of physical parameters allow for the presence of model errors and ensure the physical meanings of the identified parameters. The L-curve method for determining the regularized parameter is revised. Chebyshev polynomial is applied to approximate the acceleration response for moderation of measurement noise. Results from numerical simulations reveal that the proposed method can lead the identified physical parameter converge to a relative correct direction and it has superior performance than the traditional Tikhonov Regularization method.
(4) A new sensor placement method with multiple objectives is proposed for damage identification based on sensitivity approach from dynamic response. The covariance matrices of the identification error caused by the model errors, measurement noise and errors in the exciting forces are calculated separately. The sensor placement problem is then formulated as a multi-objective optimization problem of finding the Pareto optimal sensor combinations that compromise the criteria which are defined separately based on the covariance matrices corresponding to the three different kinds of uncertainties. A heuristic algorithm for Pareto optimal sensor placement is applied to solve the multi-objective problem. Results from numerical simulations reveal that the sensors selected from the proposed method would lead to acceptable errors of damage identification even with multi-uncertainties.
针对上述问题,本文开展了时域系统辨识问题的算法优化及传感器优化布置方法的研究。论文的主要工作和取得的成果如下：
(1)基于时域动态荷载识别方程的不适定分析,从识别方程的性态出发,提出了一种新的传感器优化布置准则——最小不适定性准则,并基于该准则提出了两种传感器数目确定条件下的位置优化方法：一种是基于结构系统马尔科夫参数矩阵条件数的直接算法,其缺点是当可能的传感器组合数目较大时,计算较为耗时：另一种是基于马尔科夫参数矩阵相关性分析的快速算法,定义了可以描述马尔科夫参数矩阵性态的相关性矩阵及传感器布置的优化指标。数值模拟结果表明,由两种传感器优化布置方法确定的最优传感器布置均可获得稳定性好、计算精度高的荷载识别结果,可用于解决时域动态荷载识别的传感器优化布置问题；随着备选传感器组合数目的增加,直接算法的计算时长会显著增加,而快速算法几乎不变,计算效率明显占优。
(2)基于转换矩阵的概念,将动态荷载识别的状态空间法拓展成了外界激励未知条件下的结构时域响应重构方法,仅利用部分测点的动态响应,通过转换矩阵重构出其他未测试位置处的响应,可用于解决时域辨识中输出数据不完备的问题。此外,还提出了一种传感器两步布设法：第一步,以重构方程具有稳定解为目标,基于单边Jacobi变换法和QR正交三角分解对全部备选测点对应的马尔科夫参数矩阵进行奇异值分解,将非零奇异值对应的传感器位置作为初始传感器布置；第二步,采用逐步积累法,以噪声效应放大指标最小为目标,在初始布置的基础上逐步增加传感器,直至达到收敛要求后获得最终传感器布置。数值模拟结果表明,该方法可根据工程实际需要,在保证重构方程具有稳定解的前提下,灵活确定最终传感器布置,获得所需的重构精度。
(3)针对振动响应灵敏度损伤识别方法,提出了一种修正Tikhonov正则化方法,可用于解决同时考虑测量噪声和模型误差干扰的条件下,传统Tikhonov正则化解不易收敛的问题。首先,对边界约束实施阈值控制,以保证解的物理意义;其次,对确定正则化参数的L-曲线方法进行修正；再次,对测量响应进行切比雪夫多项式去噪处理,减小噪声对识别结果的不利影响。数值模拟结果表明,当同时考虑噪声干扰和模型误差时,修正Tikhonov正则化方法可以使待识别的结构刚度参数逐渐收敛到一个相对正确的路径上,其损伤识别精度明显优于传统正则化方法。
(4)针对振动响应灵敏度损伤识别方法,提出了一种基于多重优化目标的传感器优化布置方法。首先,推导了结构刚度差异参数对三种典型不确定性因素——模型误差、测量噪声和荷载误差的灵敏度,进而得到不同因素所对应的识别误差协方差矩阵；然后,基于识别误差最小准则,定义了考虑多重不确定性因素的目标函数,并采用启发式搜索算法,获得了多重优化目标问题的Pareto最优解。数值模拟结果表明,考虑多重不确定性因素的条件下,由该方法确定的最优传感器布置,其损伤识别的准确性和可靠性均比较高。
Structural health monitoring and condition assessment on important civil engineering structures is an active area of research. Structural system identification including parameter identification and force identification plays an important role in structural health monitoring and condition assessment. Much attention has been devoted to this area over the last decades and various methods have been proposed, which are either in the frequency domain or time domain. Increasing interests have been focused on time domain methods in recent years, because comparing to frequency domain methods, time domain methods can directly use the measured signal and they are much easier for practical application. However, duo to the complexity of the structure and influences of environmental perturbations, there are still some problems needed to be solved, e.g. incomplete measurement data, multiple uncertainties such as noise and model errors and ill-conditioning of the identification equation etc., have adverse effects on identification accuracy. Moreover, dynamic tests should be carried out before structural system identification, and the degrees of freedom (DOFs) with sensors are limited comparing to the DOFs of the whole structure. Thus, the accuracy of system identification may vary significantly with different spatial location of the response measurements.
The methods of algorithm optimization and optimal sensor placement for structural system identification including force identification and damage identification in time domain are presented in this dissertation. The main contents and achievements are as follows:
(1) A new criterion for optimal sensor placement is presented based on the ill-conditioning analysis of the force identification equation in state space, and it is called criterion of the minimization of ill-conditioning. Two different sensor placement methods with determined number of sensor based on the proposed criterion are presented. The first one is based on direct computation of the condition number of the system Markov parameter matrix, and it would be time consuming when a large number of candidate combinations of sensor locations is considered. The second approach is based on the correlation analysis of the system Markov parameter matrix. A sensor correlation matrix is defined and the correlation criterion, which can indicate the ill-conditioning of the Markov parameter matrix, is introduced. Results from numerical simulations reveal that the performances of both methods are similar when the number of candidate combination of sensors is small. However, when there are a large number of candidate combinations, the method based on correlation analysis of the Markov parameter matrix performs better with consistently good sensor placement for force identification and much less computation effort.
(2) Dynamic force identification in state space is transformed to structural dynamic response reconstruction. The unmeasured structural responses can be reconstructed from limited measured responses. A new two-step sensor placement method is proposed for better prediction of the dynamic response reconstruction. In the first step, the system Markov parameter matrix corresponding to candidate sensor locations are singular value decomposed with One-sided Jacobi-transformation and QR decomposition methods. Sensor locations with non-zero singular values are combined as the initial sensor combination. In the second step, a measurement noise effect index is defined and the number and locations for the final sensor placement can be obtained from a heuristic forward sequential sensor placement algorithm based on the minimization of the noise effect index. Results of numerical simulations reveal that the sensors selected from the proposed method would lead to acceptable error of response reconstruction even with measurement noise.
(3) Damage identification equation based on sensitivity approach from the dynamic responses is ill-conditioned and is usually solved with regularization method. When the structural system contains measurement noise and model errors, the identification results from Tikhonov Regularization method often diverge after several iterations. A Modified Tikhonov Regularization method is presented to solve the above problem. New side conditions with limits on the identification of physical parameters allow for the presence of model errors and ensure the physical meanings of the identified parameters. The L-curve method for determining the regularized parameter is revised. Chebyshev polynomial is applied to approximate the acceleration response for moderation of measurement noise. Results from numerical simulations reveal that the proposed method can lead the identified physical parameter converge to a relative correct direction and it has superior performance than the traditional Tikhonov Regularization method.
(4) A new sensor placement method with multiple objectives is proposed for damage identification based on sensitivity approach from dynamic response. The covariance matrices of the identification error caused by the model errors, measurement noise and errors in the exciting forces are calculated separately. The sensor placement problem is then formulated as a multi-objective optimization problem of finding the Pareto optimal sensor combinations that compromise the criteria which are defined separately based on the covariance matrices corresponding to the three different kinds of uncertainties. A heuristic algorithm for Pareto optimal sensor placement is applied to solve the multi-objective problem. Results from numerical simulations reveal that the sensors selected from the proposed method would lead to acceptable errors of damage identification even with multi-uncertainties.
引文
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