In principle, List 1|]# smoothing estimates the states at time k given the measurements at a time greater than k. Most smoothing algorithms utilize forward and backward passes to find the estimates of the states at every epoch of the system output. In the popular Rauch-Tung-Strieble (RTS) smoother [6], the forward estimation is obtained using standard KF and the estimation of the backward pass is based on the maximum likelihood estimates. The main advantages of this algorithm are high reliability and simple implementation. Liu et al.[7] developed Two-Filter Smoothing (TFS) and applied it in INS/GPS integration for post-processing applications. The estimation accuracies of TFS and RTS smoother are comparable.
The computational times are similar as well.
In comparison to forward KF, the improvement of smoothing in positioning error ranges from 35% to 95% depending on the length of GPS signal outages. Chiang [8] proposed a combination of RTS smoothing and artificial neural networks (ANN) for accurate INS/GPS integrated position and orientation determination. The research illustrated that the improvement of ANN-RTS algorithm compared to RTS is about 70%. However, the extra computational time for ANN-RTS algorithm is significant due to the training process.In general, the estimation accuracy of smoothing is superior to that of filtering. However, most smoothing techniques have been applied for post-processing applications since the backward process always starts from the end of the forward filtering mission.
This limits smoothing in real-time applications.
The present Brefeldin_A study utilizes smoothing to on-line update the states of the system for near-real-time applications.2.?Optimal Estimations and Problem Statements2.1. Kalman FilterThe KF is considered as a special form of Bayesian estimation [9,10], in which the system and measurement models are originally linear or linearized into linear functions as shown:xk=��k?1;kxk?1+wk(1)zk=Hkxk+vk(2)where xk Rnx is the state vector at time k, ��k?1;k is the state transition matrix from epoch k ? 1 to k, wk Rnx is the system noise, zk Rnz is the aiding measurement, Hk is the measurement mapping matrix, and vk Rnv is measurement noise.
In the KF, Gaussian distribution is assumed for the system and measurement noise with zero mean and covariances Qk and Rk, respectively:wk~N(0,Qk)(3)vk~N(0,Rk)(4)With this assumption, the GSK-3 prior and posterior probability density function (PDF) of state vector given aiding measurements, p(xk | zk?1), p(xk | zk) are normal distribution functions.p(xk|zk?1)=N(xk;x^k|k?1,Pk|k?1)(5)p(xk|zk)=N(xk;x^k|k,Pk|k)(6)where N(xk; x?k|k, Pk|k) denotes a normal distribution of xk with mean x?k|k and covariance Pk|k.