# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 1 Introduction  
Structural integrity (diagnosis) analysis and estimation of useful life (prognosis) is carried out with the help of an array of sensors in a Structural Health Monitoring (SHM) system [1]. However, reliable diagnosis and prognosis of the health of a structural system based on input-output measurements are complicated and actively pursued by researchers around the world. With the advancement of smart materials, the focus is mainly on real-time structural monitoring, which can be evolved into condition-based maintenance philosophies [2]. Earlier work on damage identification and health monitoring of structural and mechanical systems, from changes in dynamic characteristics to the response of structures, can be found in an extensive literature survey by [PERSON] et al. [3]. An ideal structural damage detection system is capable of detecting small structural damages at an early stage, locating the damaged members and estimating the extent of the damage so that the remaining life of the structure can be estimated or mitigating measures can be undertaken. Structural damages, if not detected in time, will lead to progressive damage and will adversely affect the structural integrity. Failure to detect damages earlier also leads to increased cost of repairs. In a recent review of the structural monitoring programs implemented in the United States, [PERSON] and [PERSON] [4] have highlighted the importance of an SHM program as a powerful tool for structural damage and condition assessment of large-scale civil infrastructures after potentially damaging events. A vibration-based SHM system contains useful information about the dynamic characteristics of a system for structural damage detection. Vibration-based structural damagedetection methods are categorized into frequency domain methods and time domain methods. Frequency domain methods are mainly considered as offline methods since signals from the time domain method are processed to obtain modal information in the frequency domain. Comparison of modal information with the healthy system gives an indication of structural damage [5]. The sensitivity of the modal characteristics to the structural damages makes it difficult to identify small structural changes. All frequency domain methods are modal-based methods, and a reliable analytical model is necessary for the successful application of frequency domain methods. [PERSON] and [PERSON] [6] have drawn attention to the fact that modal-based damage detection techniques find limited real-world applications due to shortcomings stemming from environmental and operational variations and overreliance on modal-based Damage Sensitive Features (DSFs). Using measured input-output data from the vibration of a structure, entries in the system matrices are modified to reproduce the measured data [7]. Modal parameters of the damaged structure are compared with the modal parameters obtained from the analytical model of the structure, and error is minimized by iteratively refining the stiffness and mass matrices. Perturbation in mass matrices and stiffness matrices gives spatial information about damage, while quantitative analysis of the perturbation matrix can help in estimating the severity of the damage. [PERSON] [8] used the minimization of the rank of perturbation matrix to update the optimal matrix instead of using minimization of the norm of the perturbation matrix [9; 10]. [PERSON] [11] has emphasized the importance of continuous vibration monitoring together with statistical feature extraction algorithms for their application to structural health monitoring. Finite element model updating methods is one of the efficient, non-destructive and global identification techniques used. However, the main challenge for large structures that are tested in operational conditions is to identify enough modal parameters with sufficient accuracy [12]. [PERSON] et al.