# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 1 Introduction
[12] have proposed operational modal analysis with the exogeneous inputs (OMAX) approach, where an artificial force is used in operational conditions and a structural model is identified that takes both the forced and the ambient excitation into account. [PERSON] and [PERSON] [13] used experimental eigenfrequencies and mode shapes to tune the stiffness properties of the elements in the FE model update. [PERSON] et al. [14] have used a grillage method to create the adjacent precast PC box beam bridge models and used minimization of coherence function of response spectrum (CFRS) indices in the model updating method. The response spectra of two adjacent beams obtained by applying fast Fourier Transform (FFT) of either velocity or acceleration response vectors from the coupled vehicle-bridge vibration analysis model are used for CFRS index calculations. [PERSON] and [PERSON] [15] have formulated a multi-response Gaussian process (MRGP) meta-modeling approach to inversely identify model parameters through sampling in the FE model updating methodology. The response of the damaged system is different from that of the healthy system and forms the basis of the time domain methods. Structural damage detection filters are designed to detect structural damage in real-time by observing the residuals from each filter. Earlier work is attributed to failure detection, where the aim was to detect failure among sensors, actuators, and structural members and to make the failure detection algorithm robust with respect to system uncertainties and noise corrupted measurements [16]. In the course of development, fault detection based on analytical redundancy has emerged. Analytical Redundancy (AR) is based on the mathematical model of the system, where the inherent redundancy contained in the static and dynamic input-output is exploited. However, input-output measurements are coupled, and the structural changes at one location results in changes in output measurements at several locations. [PERSON] et al. [17] provided an interaction matrix formulation to decouple output error signals. [PERSON] et al. [18] detected and tracked failure among the actuators in real-time via decoupling of input-output measurements. [PERSON] [19] and [PERSON] [20] were among the first ones to implement modal-based observers as failure detection filters in the early 1970s. The second-order version of [PERSON]'s failure detection filter was developed by [PERSON] [21] and an applied filter-based approach to identify structural damage as well as the location of damage in real-time. [PERSON] et al. [22] used the Kalman filter bank approach for the damage detection method in the rotor dynamic system. In another study, [PERSON] [23] applied the extended Kalman filter with a recursive instrumental variables technique to estimate the crack development in a rotor with measured displacement in the disc. [PERSON] et al. [24] applied ARMarkov observers to estimate the severity of the damage in a planar truss structure. However, depending upon the accuracy with which structural damage needs to be estimated, a large number of ARMarkov observers are necessary for its successful application to structural health monitoring. Due to the unique input error function for each individual structural member, the proposed algorithm in this study overcomes the limitation of a large number of ARMarkov observers, system uncertainties, and noisy output measurements. Either deterministic methods or probabilistic methods are implemented to address system uncertainties. Ideally, a single set of optimal parameters that accurately represent the structural system are targeted. However, due to the complexity of the structural system with intertwined input-output measurements, probabilistic methods are preferred. Model updating based on Baysian inference is commonly used for SHM applications.