Various attempts have been made to study crack-propagation due to cyclic fatigue. The fatigue process that caused the mid-air disintegration of two de Havilland Comets in 1954 has been worked upon to develop a proper quantification theory. Engineers initially used empirical approaches like the S-N curve and the Miner’s Equation. Various approaches modified by material properties lead Paris to define a unique equation that later came to be known as Paris Law. Modifications by other prominent researchers lead to some minor modifications and classification of short cracks thus clarifying the concept of ‘short cracks’ better. However, some properties were defined empirically and this paper has tried to address the resulting problem by defining a new approach to estimate the Paris Law constants and formalize crack length evolution as a nonlinear filtering problem. **Extended** **Kalman** Filter (EKF) technique has been used here to determine the Paris Law constants. The paper has proposed using the second order Taylor expression instead of the pre-existing Mean Value First Order Second Moment (MVFOSM) approach. Future work involves implementation of the aforementioned approach and determining its computational accuracy.

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Abstract. Two data assimilation (DA) methods are compared for their ability to produce an accurate soil moisture analysis using the Météo-France land surface model: (i) SEKF, a sim- plified **extended** **Kalman** filter, which uses a climatological background-error covariance, and (ii) EnSRF, the ensemble square root filter, which uses an ensemble background-error covariance and approximates random rainfall errors stochas- tically. In situ soil moisture observations at 5 cm depth are assimilated into the surface layer and 30 cm deep observa- tions are used to evaluate the root-zone analysis on 12 sites in south-western France (SMOSMANIA network). These sites differ in terms of climate and soil texture. The two methods perform similarly and improve on the open loop. Both meth- ods suffer from incorrect linear assumptions which are par- ticularly degrading to the analysis during water-stressed con- ditions: the EnSRF by a dry bias and the SEKF by an over- sensitivity of the model Jacobian between the surface and the root-zone layers. These problems are less severe for the sites with wetter climates. A simple bias correction technique is tested on the EnSRF. Although this reduces the bias, it mod- ifies the soil moisture fluxes and suppresses the ensemble spread, which degrades the analysis performance. However, the EnSRF flow-dependent background-error covariance ev- idently captures seasonal variability in the soil moisture er- rors and should exploit planned improvements in the model physics.

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tracking of harmonic fluctuation which has the advantage of fast adaptivity for abrupt input changes. The disadvantage of KF is that it cannot be used for nonlinear systems. For nonlinear systems, the EKF has been proposed which is based on the linearization of nonlinear function using Taylor series. EKF and its variants have been used in different applications, In [15], Reggiani et al. used EKF for phase noise estimation of multi-input multi output transmission. Reif et al. [16] used EKF as a state estimation for nonlinear deterministic system and showed that the estimation error is exponentially stable. Carlos et al. [17] studied the behaviour of EKF for chaotic signals. Shmaliy [18] proposed nonlinear **extended** finite impulse response **filters** which does not require the noise statistics and initial error. EKF has the disadvantage of large linearization error which may lead in wrong estimations. To reduce the linearization error, IEKF has been proposed in the literature and used for various applications [19]-[22]. The paper is organized as follows. Section II gives brief theory of EKF. Section III presents IEKF method. Section IV presents state modeling of CE amplifier circuit using Kirchhoff’s law and transistor model. Section V presents implementation of EKF and IEKF to CE amplifier circuit. Section VI presents simulation results. Section VI concludes the paper.

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systems. But when the system is suffering with Gaussianity then only it will works. While coming for the non-gaussianity it will not estimate the state efficiently. Hence, it is also to all the applications of classical **filters**. As a remedy of the both **Kalman** filter and **extended** **Kalman** **filters** there is one more classical filter named as unscented **Kalman** filter. Which is classical filter mostly works for state estimation of the non-linear systems having non-Gaussianity feature. By using this unscented **Kalman** filtering, most of the applications may executed or estimated while coming for the estimation of the states, it held efficiently in stable and stored information only. Means it will not give the efficient estimation of states for online estimation (will not be able to operate/ calculate efficient state estimation). We may consider this as a drawback because of now-a-days time and storage are the basic considerations. If we are capable to do in online then there is no need of storage and most of the time going to be save. Hence, then as a remedy of all these problems one non-classical filter is invented which is named as particle filter.

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The widely used EKF is based on linearized approximations to nonlinear dynamic and/or measurement models. For this case, the linearized approximation is performed in the measurement update step. The **extended** **Kalman** filter extends the scope of **Kalman** filter to nonlinear optimal filtering problems by forming a Gaussian approximation to the joint distribution of state x and measurements y using a Taylor series based transformation. First order **extended** **Kalman** **filters** are presented, which are based on linear and quadratic approximations to the transformation. Higher order **filters** are also possible, but not presented here. The filtering model used in the EKF is

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Recently, model-based state estimating methods are widely applied into SOC estimation. The accuracy of such methods highly depends on the battery model and the observer. The frequently utilized battery models include physics-based models. References [19], [20], equivalent circuit models [21], [22], machine learning models [23], [24], and empirical models [7]. The design of observers to estimate the state can be in different ways, such as **Kalman** filter family [25], [26], sliding observer [10], [24], H-infinity observer [27], [28], and Luenberger observer [29], [30]. Among all the kinds of observers, **Kalman** filter family takes up the largest percentage. **Extended** **Kalman** filter (EKF) was introduced to estimate SOC of a lithium-ion polymer battery by Plett [31] in 2004. Later, sigma-point **Kalman** filter was proposed to improve estimation accuracy [32]. Simultaneously, the methodologies to enhance the **Kalman** filter family’s performance on SOC estimation emerged, such

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Target tracking deals with the estimation of unknown states, such as position, velocity and acceleration of a moving tar- get, using noisy measurements in a given coordinate space. Algorithms that can accurately track the mobility of a target offer numerous advantages in a wide range of applications. The most obvious example is tracking an aircraft using radar measurements. It is of immense importance in many military applications and is also essential for air traffic control required by civilian airlines. Some other examples include tracking of a mobile node in a cellular network which is required for effi- cient radio resource management and tracking in autonomous cars and robots. Target tracking algorithms, despite having a diverse range of applications, employ a common structure based on the Bayesian filtering framework for extracting useful information from the available data. The standard Bayesian filtering solutions such as traditional **Kalman** **filters** (in case of linear systems) and sigma-point **filters** (e.g., Cubature **Kalman** Filter (CKF), Unscented **Kalman** Filter (UKF), etc., for nonlinear systems), assume that the noises have Gaussian distribution [1]. In practice, however, large deviations (outliers) occur in real data frequently and these cannot be modeled accurately by a Gaussian distribution only [2]. As a result,

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of the system equation, a sub-optimal state estimate for a nonlinear state-space system can be given in [3]. In order to reduce error in the estimation of the system state and improve the precision of EKF, various modified algorithms have been established. For example, Uhlmann et al. [4] give a kind of unscented **Kalman** filter (UKF) algorithm for nonlinear systems. This algorithm has the same calculation capacity as the EKF algorithm and re- duces error in linearizing nonlinear model. Merwe et al. [5] present an algorithm called central difference Kal- man filter (CDKF) and deduce its recursive formula. This algorithm is easier to realize than EKF, because it does not need to compute Jacobi matrices. But UKF and CDKF are not suitable for non-Gaussian state-space systems. If a state-space model is nonlinear and noise distributions are non-Gaussian, particle **filters** (PFs) or sequential Monte Carlo (SMC) methods provide an effective algorithm to deal with this case [6]. A combination of EKF and PF leads to **extended** **Kalman** particle filter (EKPF) algorithm, where EKF algorithm updates the sampling particles and the particles approximate the filtering distributions. The EKPF algorithm has been ap- plied to neural networks in [7], and the results show that the algorithm is superior to other basic particle filter algorithms. Although these algorithms have some advantages in effectiveness and calculation speed for most of nonlinear state-space models, sometimes we need some more accurate **Kalman** filter algorithms to estimate the state of nonlinear systems.

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The SISO **Kalman** Basic Controller (KBC) uses the **Kalman** filter to track the utili- sation and update the allocation accordingly. Since first presented by R.E. **Kalman** in his seminal 1960’s paper [**Kalman** 1960], the **Kalman** filter has been used in a large number of areas including autonomous or assisted navigation, interactive computer graphics and motion prediction. It is a data filtering method that estimates the state of a linear stochastic system in a recursive manner based on noisy measurements. The **Kalman** filter is optimal in the sum squared error sense under the assumptions that the system is described by a linear model, and the process and measurement noise are white and Gaussian. It is also computationally attractive, due to its recursive compu- tation, since the production of the next estimate only requires the updated measure- ments and the previous predictions; for a more comprehensive analysis of the **Kalman** filter refer to for example [Simon 2006; Maybeck 1979; Welch and Bishop 1995].

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We are indebted to a reviewer who called our attention to other recent works in which particle **filters** were applied to the acoustic source localization problem [10, 11]. As ex- plained in the tutorial by Arulampalam et al. [12], particle **filters** represent a generalization of **Kalman** **filters** that can handle nonlinear and non-Gaussian state estimation prob- lems. This is certainly a desirable characteristic, and makes particle **filters** of interest for future study. It remains to be seen, however, whether particle **filters** will prove better-suited for acoustic source localization than the **extended** **Kalman** fil- ters considered here. To wit, Arulampalam et al. [12] discuss several problems that can arise with the use of particle fil- ters, namely, degeneracy and sample impoverishment. While solutions for circumventing these problems have appeared in the literature, the application of a particle filter to a track- ing problem clearly requires a certain amount of engineer- ing to obtain a working system, much as with our approach based on the **Kalman** filter. Moreover, it is not clear that the assumptions inherent in the **Kalman** filter, namely, linearity and Gaussianity, make it unsuitable for the speaker track- ing problem: Hahn and Tretter [13] show that the observa- tion noise encountered in time delay of arrival estimation is in fact Gaussian, as required by a **Kalman** filter. More- over, as shown here, the nonlinearity seen in the acoustic source localization problem is relatively mild and can be ad- equately handled by performing several local iterations for each time step as explained in [14]. Such theoretical consid- erations, notwithstanding, the question of whether **Kalman** or particle **filters** are better suited for speaker tracking, will only be answered by empirical studies. We believe that such studies should be conducted on real, rather than simulated, data such as we have used for the experiments discussed in Section 5, as only results obtained on real data will be truly compelling. We hope to make such empirical comparisons the topic of a future publication.

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Fig. 6 yields a faster convergence of the activation times of the UKF for all measurement noise levels. The signal and process functions h and f are nonlinear and even non-smooth functions of the activation times. The EKF is an algorithm of 1st order accuracy while the UKF captures the mean and covariance of the Gaussian distributed state vector to the 3rd order Taylor series (Wan and van der Merwe, 2000). In the next steps the heart model needs to be **extended** to the total ventricles and a muscle fibre model needs to be introduced to include an anisotropic activation wavefront velocity with respect to muscle fibre orientations. For an application to re- alistic data the heart model parameters, e.g. the process noise covariance Q, need to be estimated from the data. This can be done e.g. with the Expectation-Maximization (EM) algo- rithm (Khan and Dutt, 2007).

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Power system state estimation requires error less data to estimate the exact states of the power system. The Estimation process is done by Energy Management System (EMS) at the control centre with the help of estimated data. In practical conditions, collected data contain the measurement and process errors. These errors are due to high speed measuring devices and Phasor Measurement Units (PMU) installed on different buses. Due to communication errors, different filtration techniques are required at the control centre to get the best estimated data. For nonlinear power system, new improved **Kalman** filter techniques are introduced in this paper. Emerging **Extended** **Kalman** Filter (E-EKF) and Emerging Unscented **Kalman** Filter (E-UKF) based on the exponential description function are proposed in this paper. The effectiveness of these improved techniques is compared with the conventional nonlinear filterson the basis of elapsed time and Root Mean Square Error (RMSE). The performance of these **filters** is tested on standard IEEE-30 bus test system.

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To make the test standard and avoid contamination due to human error, instead of run- ning the wheelchair over the track separately for different algorithms, the wheelchair was driven once for each speed setting mentioned above, and the data from LiDAR, Kinect, IMU, and encoders was recorded. To generate a ground truth trajectory, dead reckon- ing was done using an **Extended** **Kalman** Filter (EKF) to fuse the data from IMU and encoders 5.13 (b). This recorded data was replayed for each SLAM algorithm and a trajec- tories were generated for each. An example trajectory for Gmapping and VINS-Mono is shown in Fig. 5.14. Absolute trajectory error and Relative Pose Error for translation and Rotation was computed against the ground truth trajectory and the result is reported in tables 5.1, 5.2, and 5.3 and their corresponding column charts in Figures 5.15, 5.16, and 5.17.

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The vehicle sideslip angle is an important physical variable strongly linked to the directional behaviour and stability of the vehicle. As a consequence, the knowledge of the sideslip angle is requested from the vehicle dynamics control systems that establish their intervention on the basis of a difference between a target and a current value [1]. The measurement of the vehicle sideslip angle can be obtained by means of devices that are very expensive and not functional for an easy installation on the car. The well- known industrial solutions typically rely on observers, which are based on heavily simplified dynamic vehicle models in combination with kinematic models. Methodologies for vehicle sideslip angle estimation can be found in literature. Many of them are generally based on the **extended** **Kalman** filter (EKF) algorithm [2]. In any case, the nonlinear nature of the vehicle system strongly limits the performance of linear and linearization based approaches [3] that inevitably give estimation errors that affect the performance of the vehicle dynamics controllers employing the estimated variables as feedback. In order to overcome this limit, this paper investigates on a nonlinear technique

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tightly coupled GPS/INS navigation systems based on extended and sigma point Kalman. 16[r]

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Previously, subspace identification methods (van Over- schee and de Moor 1996) achieved significant advances in the learning of LDS. For a sequence of observations gen- erated from an LDS, this family of methods allows to re- cover the state sequence of the **Kalman** filter of the LDS via a singular value decomposition (SVD) of a certain ma- trix constructed from the inputs. In a naive implementation, this requires the SVD to be performed at each time step, on a matrix constructed from the full history of observations. (Venkatraman et al. 2016) proposed an on-line method re- lated to subspace identification using the notion of instru- mental variables. While the experimental part of the paper deals with LDS forecasting, the provided theoretical guaran- tees apply only in cases of independent observations, which is not the case for LDSs. More broadly, guarantees we are aware of require that the observations are generated from an LDS that is stationary, in contrast to finding the optimal fil- ter for a given arbitrary sequence. This therefore excludes most tracking applications.

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From an approximation of the a posteriori density func- tions of the data signals by a weighted sum of Gaussian den- sity functions and by exploiting the mixture model of the observation noise, we propose here a new robust structure of a multiuser detector that is based on a network of **kalman** **filters** operating in parallel. Under the common MMSE esti- mation error criterion, the state vector (consisting of the last transmitted symbols of all users) is estimated from the re- ceived signal, where **Kalman** parameters are adjusted using one noise parameter (variance and contamination constant) and one Gaussian term in the a posteriori pdf approxima- tion of the plant noise. This version is called **extended** NKF detector.

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the vehicle and it is often considered to increase the performance of the car as well as to develop safe- ty systems, especially in the vehicle equipped with Torque Vectoring control systems. This paper de- scribes two methods, based on the use of **Kalman** **filters**, to estimate the vehicle sideslip angle and the tire forces of a vehicle starting from the longitudinal and yaw velocity data. In particular, these data re- fer to on-track testing of a Range Rover Evoque performing ramp steer maneuvers at constant speed. The results of the sideslip estimation method are compared with the actual vehicle sideslip measured by a Datron sensor and are also used to estimate the tire lateral forces.

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This work presents a solution for static hysteresis func- tion identification with a multiple model approach. The hysteresis function is described by a set of elementary models connected in parallel. This parallel structure of the model, in combination with the IMM estimator gives the possibility to reduce the nonlinear identification problem to a linear one. For the parameters (the spring constant and the force coefficient) of each elementary model, grids of possible values are preset taking into account physical restrictions. With each elementary model a separate IMM estimator is synthesized working by linear **Kalman** **filters** based on models with different parameters. The final estimate of each parameter represents a fusion of the values from the grid weighted by the IMM mode probabilities. The estimated output of each elementary model is a fusion of the weighted estimates of the **Kalman** **filters** by the probabilities, and the total hysteresis force represents the sum of the estimated outputs of all elementary models. A compari- son with the recursive least-squares method is discussed.

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Abstract-- An efficient moving object segmentation algorithm suitable for real-time content-based multimedia communication systems is proposed in this paper. First, a background registration technique is used to construct a reliable background image from the accumulated frame difference information. The moving object region is then separated from the background region by comparing the current frame with the constructed background image. Finally, a post-processing step is applied on the obtained object mask to remove noise regions and to smooth the object boundary. Surveillance system can be used to detect and track the moving objects. First phase of the system is to detect the moving objects in the video and track the detected object. Second phase of the system detected different abnormal activities like crimes and robbery in ATM. In this paper, detection of the moving object has been done using simple background subtraction and tracking of single moving object has been done using **Extended** **Kalman** filter. Detection of abnormal activities can be done by using HOG (Histogram of Gradient) and IM (Illumination Mapping). The algorithm has been applied successfully on standard surveillance video datasets. The proposed method will uses multiple object detection method and event recognition techniques of computer vision.

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