Summary We provide a tutorial for a number of variants of the extended Kalman filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, unscented Kalman filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is central difference Kalman filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling's polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and square root central difference Kalman filter (SR‐CDKF). These SR‐versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step‐by‐step algorithms of above‐mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both mean squared error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR‐CDKF show the best MSE performance in most scenarios; particularly, SR‐CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR‐CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR‐CDKF taking advantage of its estimating performance and high processing speed. Copyright © 2016 John Wiley & Sons, Ltd.
Passive sonar systems are used to detect the acoustic signals that are radiated from marine objects (e.g., surface ships, submarines, etc.), and an accurate estimation of the frequency components is crucial to the target detection. In this paper, we introduce sparse Bayesian learning (SBL) for the frequency analysis after the corresponding linear system is established. Many algorithms, such as fast Fourier transform (FFT), estimate signal parameters via rotational invariance techniques (ESPRIT), and multiple signal classification (RMUSIC) has been proposed for frequency detection. However, these algorithms have limitations of low estimation resolution by insufficient signal length (FFT), required knowledge of the signal frequency component number, and performance degradation at low signal to noise ratio (ESPRIT and RMUSIC). The SBL, which reconstructs a sparse solution from the linear system using the Bayesian framework, has an advantage in frequency detection owing to high resolution from the solution sparsity. Furthermore, in order to improve the robustness of the SBL-based frequency analysis, we exploit multiple measurements over time and space domains that share common frequency components. We compare the estimation results from FFT, ESPRIT, RMUSIC, and SBL using synthetic data, which displays the superior performance of the SBL that has lower estimation errors with a higher recovery ratio. We also apply the SBL to the in-situ data with other schemes and the frequency components from the SBL are revealed as the most effective. In particular, the SBL estimation is remarkably enhanced by the multiple measurements from both space and time domains owing to remaining consistent signal frequency components while diminishing random noise frequency components.
Accurate estimation of the frequency component is an important issue to identify and track marine objects (e.g., surface ship, submarine, etc.). In general, a passive sonar system consists of a sensor array, and each sensor receives data that have common information of the target signal. In this paper, we consider multiple-measurement sparse Bayesian learning (MM-SBL), which reconstructs sparse solutions in a linear system using Bayesian frameworks, to detect the common frequency components received by each sensor. In addition, the direction of arrival estimation was performed on each detected common frequency component using the MM-SBL based on beamforming. The azimuth for each common frequency component was confirmed in the frequency-azimuth plot, through which we identified the target. In addition, we perform target tracking using the target detection results along time, which are derived from the sum of the signal spectrum at the azimuth angle. The performance of the MM-SBL and the conventional target detection method based on energy detection were compared using in-situ data measured near the Korean peninsula, where MM-SBL displays superior detection performance and high-resolution results.
We consider a model that minimizes the total cost incurred by assigning available weapons to existing targets in order to reduce enemy threats, which is called the weapon target assignment problem (WTAP). This study addresses the stochastic versions of WTAP, in which data, such as the probability of destroying a target, are given randomly (i.e., data are identified with certain probability distributions). For each type of random data or parameter, we provide a stochastic optimization model on the basis of the expected value or scenario enumeration. In particular, when the probabilities of destroying targets depending on weapons are stochastic, we present a stochastic programming formulation with a simple recourse. We show that the stochastic model can be transformed into a deterministic equivalent mixed integer programming model under a certain discrete probability distribution of randomness. We solve the stochastic model to obtain an optimal solution via the mixed integer programming model and compare this solution with that of the deterministic model.
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