Nonlinear system identification for vibration-based structural health monitoring

A monitoring approach is adopted that consists of three stages: local modal parameter identification, iden tification of a global system model using the modal parameters as outputs, and monitoring the prediction error. Given the need for effective nonlinear global models in structural health monitoring, the performance of two kernel-based sys tem identification methods on real-life monitoring data is investigated: kernel PCA for output-only monitoring and least-squares SVM for input-output monitoring. Both approaches perform very well for the validation case considered, and they can improve the current practice in structural health monitoring in terms of robustness, computation effort and prediction accuracy. Keywords— Structural health monitoring, operational modal analysis, nonlinear system identification I. PROBLEM STATEMENT ONE of the major issues in system monitoring applications is that unwanted changes in the system behavior, due to defects for example, may be of the same order of magnitude as, or even completely hidden by, normal variations in its parameters. In this work, a special case of this very general problem is considered, where (1) the system dynamics are linear and time-invariant for fixed parameter values, and (2) the parameter variations are slow in the sense that they are band-limited, with an upper frequency limit that is much smaller than the lowest eigenfrequency obtained for any admissible set of fixed parameter values. A prime example where this case applies is vibrationbased structural health monitoring (SHM) under changing weather conditions: the stiffness of structural materi ls depends on the temperature, but the variations in outdoor air temperature are much slower than the lowest structural eigenperiod for a fixed temperature value. When monitored for a short period of time (seconds, minutes), the structure therefore behaves like a linear time-invariant s ystem, but these local linear dynamics change when monitored over longer time spans (hours, days, months, years). In terms of the underlying cause (changes in outdoor air temperature), these changes are in principle nonlinear, due to the nonlinear temperature-stiffness relationships of structural materials, and dynamic, due to the large thermal inertia of most structures. In this paper, the following methodology for monitoring time-varying systems is chosen. (1) Data reduction by breaking up the available training data in time sequences that are short compared to the parameter variations, identifying a local linear time-invariant system model for each individual sequence, and computing the corresponding modal parameters; (2) Identification of a global (possibly nonlinear and dynamic) system model, where the timevarying modal parameters of the first step are considered as outputs. (3) Monitoring the system by continuously repeating step 1, and comparing the modal parameters found in this way with the values that are predicted by the model resulting from step 2. Changes in the system behavior that were not present in the training data, will cause a growth in the prediction error, which makes them detectable. Although during the last decades, research on vibrationbased SHM has been expanding rapidly [1], [2], the nonlinearity relationship between temperature and stiffness is often neglected. Peeters and De Roeck [3] report a bridge vibration monitoring study where the relationship can be linearized to fair accuracy for a limited temperature range only. They neglected the monitoring data outside this range and applied linear system identification on the remaining data, thereby stating that an extension would be desirable so that the structure can be monitored also when t outdoor air temperature falls outside the considered range. Ni et al. [4] introduced nonlinear modelling in this context by using support vector machines (SVMs) for identifying a model relating local eigenfrequency values obtained from measured short-term acceleration sequences (outputs), to measured temperatures (inputs). The performance of the identified model was not satisfactory, and it was later improved by performing a preliminary linear principal component analysis (PCA) of the 83 temperature signals, retaining the first 16 principal components, and us ing these as inputs to the algorithm [5]. In both cases, the static models were identified by means of quadratic programming; no validation could be performed as no damge occurred during the monitoring period. Yan et al. [6] considered output-only modelling of eigenfrequency variations by assuming a piecewise linear relationship between the outputs, and applying linear PCA for each piece. However, this approach has limited applicability and it was found that the parameter values involved should be chosen with great care in order to yield useful results.