## 2 Background: Identification of Dynamic Systems by Unscented Kalman Filter
### Applications of Filter Methods and Practical Issues  
In applications of filter methods to structural systems, it is generally assumed that the accelerations can be measured. For the purpose of _state_ estimation, _parameter_ estimation, or _input_ estimation, the corresponding state vector is respectively defined in terms of the dynamic responses (e.g., displacement and velocity), system parameters (e.g., Young's modulus, damping coefficient, and parameters of Bouc-Wen hysteretic models), or input forces. If system parameters are estimated concurrently with dynamic responses, the filter is termed as _joint state-parameter_ estimation. In the so-called _joint input-state_ estimation, input forces are estimated together with the responses. For example, as a joint state-parameter estimation, [PERSON] et al. (2006) proposed an adaptive EKF-based tracking technique to detect sudden damage in a five-story linear structure [11]. [PERSON] and [PERSON] (2007) applied UKF to a nonlinear system having 2 degrees of freedom (DOFs) for the purpose of SI to find that the UKF provided better estimation than EKF for highly nonlinear systems [21]. [PERSON] and [PERSON] (2009) also utilized UKF for SI of nonlinear 3-DOF structural system assuming non-collocated heterogeneous sensing [22]. Most previous studies on joint state-parameter estimation have focused on improving performance of filters in tracking parameters of highly nonlinear structural system. These studies employed the system equation of motion under the assumption that either acceleration or displacement is measured at every DOF to avoid rank-deficiency. However, measuring dynamic responses at all DOFs is infeasible especially for structures consisting of many components. This may limit the applicability of joint state-parameter estimation to SI in practice. Recently, many research efforts have been made regarding the joint input-state estimation to track the input force whose exact information is not available in many situations. [PERSON] and [PERSON] (2007) proposed the recursive-three-step filter (RTSF), which updates responses and input force at the different step, for unbiased minimum-variance estimation of input and state [23]. Since RTSF suffered from rank-deficiency caused by a limited number of measured acceleration or reduced-order model from modal analysis, [PERSON] et al. (2012) proposed an advanced version of RTSF by using truncated covariance matrices through singular value decomposition [24]. [PERSON] et al. (2015) developed a dual Kalman filter consisting of two stages to prevent numerical issues such as un-observability and rank-deficiency of RTSF when using sparse acceleration measurements [25]. [PERSON] et al. (2019) proposed an augmented Kalman filter, whose state vector consists of responses and input together, with the state-space model developed based on the generalized-\(\alpha\) method [18]. This was to address the issue that the dual Kalman filter is sensitive to selection of covariance matrices of artificial noises. These studies focused on joint input-state estimation of linear structural systems with time-invariant system parameters. Although full-measurements are not required because of the linear model assumption, rank-deficiency and divergence are still critical issues when SI relies on sparse measurements or a reduced-order model is used. Meanwhile, as a parameter estimation example, [PERSON] et al. (2019) applied UKF to Finite-Element (FE) updating with adaptive techniques to consider uncertainty of nonlinear FE models [13]. The state-space model based on Newmark method was able to utilize sparse measurements whereas the use of equilibrium equations in state-space models of previous joint state-parameter estimations needs full measurements. Then, dual adaptive filtering was performed based on the proposed state-space model to consider uncertainty in FE model.