## 3 Proposed Method: Adaptive Unscented Kalman Filter Using Sparse Measurements of Accelerations
### Dual Adaptive Filtering for Measurement Noise Estimation  
For accurate estimation of time-variant system parameters, the advanced state-space model needs to be used in conjunction with a proper adaptive filtering technique. There are two reasons for the limited tracking ability of conventional filters when statistics in measurements change. First, the predicted state vector in the time-update part is influenced by the size of covariance of artificial process noise. Second, both covariance matrices of artificial process and measurement noises affect the calculation of the Kalman gain. Therefore, if both covariance matrices are constant during estimation, the tracking performance of conventional filters will be limited for time-variant dynamic system. While developing the proposed filter, a few other adaptive techniques were explored. For example, [PERSON] et al. (2005) proposed an adaptive tracking technique for the purpose of leak detection and location in the management of a pipeline system [17]. This tracking technique was performed based on particle filter, which is also known as a sequential Monte Carlo method. However, it is noted that the idea of adaptive technique is utilized also in the UKF-based approaches. The details and limitations of this technique (with modification for UKF) are described in Appendix A. It is especially noted that this adaptive technique controls the artificial process noise, which causes severe bias and instability of estimations. The other adaptive technique explored in this study is a master-slave Kalman filter (MS-KF), which is also called as a dual adaptive filtering. The main concept of MS-KF is that, when a master filter (MF) estimates the given random variables, the slave filter (SF) will estimate the covariance matrix of either artificial process or measurement noises in MF at every time-step. There are two types of MS-KF based on process noise estimation (MS-KF\({}_{\rm PE}\)) and measurement noise estimation (MS-KF\({}_{\rm ME}\)). Among these, details of MS-KF\({}_{\rm PE}\) are briefly presented in Appendix B because MS-KF\({}_{\rm PE}\) is not suitable for the goal of this study for the same reason as the adaptive technique described in Appendix A. Unlike the two techniques mentioned above, MS-KF\({}_{\rm ME}\) controls artificial measurements noise and can be utilized with the advanced state-space model. The general algorithm of MS-KF is illustrated by a block diagram in Figure 2. In the slave filter (SF) part, two random variables \(\mathbf{\omega}_{k}\in\mathbb{R}^{n_{x}\times 1}\) and \(\mathbf{v}_{k}\in\mathbb{R}^{n_{x}\times 1}\) represent the artificial noises for process and measurements respectively modeled as \(\omega_{k}\sim N(0,\mathbf{T})\) and \(\
u_{k}\sim N(0,\mathbf{U})\) in which \(\mathbf{T}\in\mathbb{R}^{n_{x}\times n_{x}}\) and \(\mathbf{U}\in\mathbb{R}^{n_{x}\times n_{z}}\) are time-invariant covariance matrices; and the superscript \(S\) indicates a variable related to the SF. A well-known drawback of MS-KF is its sensitivity with respect to the choice of two covariance matrices of artificial noises in SF. Therefore, these artificial noises should be selected carefully with those of MF [15; 27]. Linear KF (LKF) is utilized for SF in most cases because there is no accurate knowledge available for the state-space model [15]. In this paper, a covariance-matching technique is utilized in the state-space model of SF because this intuitive technique can be implemented easily with low computational costs [13; 15; 27; 33]. In Figure 2, using the equations related to the number (2) indicates that the MS-KF estimates the covariance of artificial measurements noise at every time-step of MF [13; 15] (if the equations noted by the number (1) are used, MS-KF\({}_{\rm PE}\) is performed). MS-KF\({}_{\rm ME}\) can be conducted based on two types: \"the innovation and its covariance\" or \"the residual and its covariance\". In this paper, the latter one is selected because its calculations are less complex.