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Power Systems State Estimation
Published in John Fuller, Pamela Obiomon, Samir I. Abood, Power System Operation, Utilization, and Control, 2023
John Fuller, Pamela Obiomon, Samir I. Abood
The UKF approach is comparable to the EKF method. It is, nevertheless, recognized as the “derivative-free” approach. In contrast to the linearization of the EKF, the UKF employs a non-linear function. It then obtains the state estimation by approximating the non-linear system’s probability distribution. The Unscented Transform is a powerful mathematical function that has been created to augment the Kalman Filter (U.T.). U.T. is a mathematical or statistical procedure used to transform non-linear functions into a probability distribution matching a finite set. Those altered probability points are referred to as sigma points. Following the transformation, an estimate of the outcomes is generated in the mean vector and associated error covariance matrix by applying the known non-linear function h(x) to each vector.
Kalman Filtering
Published in Chingiz Hajiyev, Halil Ersin Soken, Fault Tolerant Attitude Estimation for Small Satellites, 2020
Chingiz Hajiyev, Halil Ersin Soken
To implement the Kalman filter for nonlinear systems without any linearization step, the unscented transform and so UKF can be used. The UKF algorithm is a more recent filtering method which has many advantages over the well-known EKF. The essence of the UKF is the fact that a nonlinear distribution can be approximated more easily than a nonlinear function or transformation. Instead of the analytical linearization used in EKF, UKF statistically captures the system’s nonlinearities. The UKF avoids the linearization step of the EKF by introducing sigma points to catch higher order statistic of the system. As a result it satisfies both better estimation accuracy and convergence characteristic. The EKF is more sensitive to measurement faults than the UKF, especially in case of additional bias and noise in the measurements.
Graphical Models
Published in Stephen Marsland, Machine Learning, 2014
The Extended Kalman Filter is not an optimal estimator. Further, if the assumptions about local linearity that underlie the linearisation are not true, then the estimate is very poor, and even where it is good, it requires the computation of the Jacobians, which is potentially difficult. There have been various attempts to improve upon it; one method that may be of interest is to choose a set of points that represent the statistics of the data and to transform them by passing them through the non-linear functions (f(·) and h(·)), and then to compute the statistics of those points in order to estimate the statistics of the transformed data. This has the great name of the unscented transform, and can be used to produce an Unscented Kalman Filter. For more information on this, see the Further Reading section; instead we will look at a common MCMC algorithm for performing tracking, the particle filter.
Acceptance Criteria of Bearings-only Passive Target Tracking Solution
Published in IETE Journal of Research, 2023
S. Koteswara Rao, M. Kavitha Lakshmi, Kausar Jahan, G. Naga Divya, B. Omkar Lakshmi Jagan
The problem in EKF is also reduced by introducing the covariance matrix of the state vector in the modified gain function and this EKF is called modified gain bearings-only extended Kalman filter (MGBEKF) [17]. The modified gain of MGBEKF is developed based on the condition that nonlinearity in the system designed (relation between the state and measurement which is nonlinear) is modifiable and this algorithm was proved to be unbiased and to converge very rapidly. Likewise, many new estimation algorithms for BOT-TMA are suggested. Amongst them, UKF is of more significance [18–21]. With Unscented Transform (UT), the information is propagated through a nonlinear function, in the form of the mean of the state vector and its covariance matrix. This UT is combined with some sections of the optimal linear Kalman filter and is called UKF. UKF provides an adequately precise result for tracking applications.
Filtering based multi-sensor data fusion algorithm for a reliable unmanned surface vehicle navigation
Published in Journal of Marine Engineering & Technology, 2023
Wenwen Liu, Yuanchang Liu, Richard Bucknall
From the start of USV operation, the on-board IMU starts to measure the motions of the USV, that is, the accelerometer measures the accelerations and the gyroscope measures the angular velocity of the USV. Normally, acceleration rates provided by the IMU are along the inertial frame, which can be approximated as the body frame; whereas, other navigation information has been presented in the navigation frame. It therefore should convert the IMU data from the inertial frame to the navigation frame by using the rotation matrix: As shown in Figure 2, the heading that can be obtained from the compass is the clockwise angle referenced to North. Therefore, the anti-clockwise rotation angle from inertial frame (i-frame) to navigation frame (n-frame) is equal to the heading: It can be observed that the conversion of the frames generates the non-linearity of the system. Unscented Kalman filtering, uses an unscented transform to propagate designed Sigma points and calculates the mean of the propagated point to compute the optimal estimation of the input data. It has been used increasingly in vehicle navigation in recent years (Ma et al. 2014; Meng et al. 2016). As stated previously, when the frame rotation angle is equal to the heading of the USV, the non-linear dynamic model can then be obtained by combining Equation (6) and Equations (1) to (3) as below:
Incremental unscented Kalman filter for real-time traffic estimation on motorways using multi-source data
Published in Transportmetrica A: Transport Science, 2022
Xuan-Sy Trinh, Dong Ngoduy, Mehdi Keyvan-Ekbatani, Blair Robertson
The Unscented Kalman Filter, which proposed by Julier and Uhlmann (1997, 2004), relies on a technique called the Unscented Transform (UT). This is a deterministic sampling approach that approximates the population characteristics of a random variable by selecting a set of sample points known as sigma points. If a random variable has a known mean and covariance, its distribution can be represented by a set of sigma points and their weights which are generated through the UT. When these sigma points propagate through nonlinear functions, they capture the posterior mean and covariance accurately to the third order (Taylor series expansion) (Wan and Van Der Merwe 2000). Assuming that an n-dimensional vector has mean and covariance matrix , a set of 2n + 1 sigma points can be generated as: where denotes the ath sigma point, α, β, κ are parameters that can be chosen (see Wan and Van Der Merwe 2000; Julier and Uhlmann 2004 for how to select them). is the ath column of the square root of the covariance matrix, and is the weight that is associated with the ath point. The m and c superscripts of denote the mean and covariance weights, respectively.