Linear Observation Equations Research Articles

Iterative UKF under generalized maximum correntropy criterion for intermittent observation systems with complex non-Gaussian noise

The traditional unscented Kalman filters (UKFs) under the maximum correntropy criterion provide a powerful tool for nonlinear state estimation with heavy-tailed non-Gaussian noise. Nevertheless, the above-mentioned filters may yield biased estimates because the Gaussian kernel function can only handle certain types of non-Gaussian noise. Additionally, the use of statistical linearization methods can result in approximation errors when solving linear observation equations, while the system may also experience observation data loss. Therefore, a new iterative UKF with intermittent observations under the generalized maximum correntropy criterion is proposed for systems with complex non-Gaussian noise, called GMCC-IO-IUKF. Firstly, the connection between the UKF with and without intermittent observations is established by designing a coefficient matrix including intermittent observation variables, so as to derive the UKF with intermittent observations under the maximum correntropy criterion. Secondly, for the measurement update of GMCC-IO-IUKF, a nonlinear regression augmented model that can deal with both prediction and observation errors is established using the coefficient matrix and the nonlinear function. To better adapt to different types of non-Gaussian noise, the generalized Gaussian kernel function is substituted for the traditional Gaussian kernel function. Theoretically, GMCC-IO-IUKF can achieve better estimation performance by directly employing the nonlinear function and the latest iteration value. Finally, a classical target tracking model is used to evaluate the estimation performance and feasibility of our proposed GMCC-IO-IUKF algorithm. It appears from the experiment results that our proposed GMCC-IO-IUKF can not only promote estimation precision but also handle complex non-Gaussian noise flexibly.

Modified method of invariant immersion in the synthesis of measuring procedures for estimating the motion parameters of a maneuvering target

The problem of processing the results of radar measurements of the target motion parameters under conditions of disturbances due to aircraft maneuvers is considered. In order to adapt to the disturbances of measuring procedures, the method of invariant immersion has been modified. The application of the modified method in the case of a linear observation equation leads to a widespread form of a discrete measurement procedure obtained in this paper as a special case of modified invariant immersion equations. On the basis of statistical modeling of digital processing of the results of radar measurements of the movement parameters of the maneuvering target, the efficiency indicators of radar stations are calculated – these are the averaged absolute and relative errors in the measurement of motion parameters, the averaged wiring coefficient and the averaged time of the duration of the breaks of the tracks. Their analysis allows us to assert the high efficiency of using new measuring procedures synthesized using acceleration models in the form of white noise and Singer in the information and measurement systems of radar stations. This becomes possible due to the presence of the matrix coefficient of adaptation in the structure of the modified equations of invariant immersion. The comparison was carried out with Kalman measurement procedures synthesized using these acceleration models. Calculations of performance indicators confirm the reliability of the results obtained, since the wellknown fact of an increase in the error of measuring the height of an aircraft by monopulse radar stations near the surface is demonstrated. The results of the study will be useful in the development of adaptive measurement procedures for information and measurement systems operating under conditions of disturbances of measuring processes, for example, for survey locators.

Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation

Observational errors of Particle Filtering are studied over the case of a state-space model with a linear observation equation. In this study, the observational errors are estimated prior to the upcoming observations. This action is added to the basic algorithm of the filter as a new step for the acquisition of the state estimations. This intervention is useful in the presence of missing data problems mainly, as well as sample tracking for impoverishment issues. It applies theory of Homogeneous and Non-Homogeneous closed Markov Systems to the study of particle distribution over the state domain and, thus, lays the foundations for the employment of stochastic control against impoverishment. A simulating example is quoted to demonstrate the effectiveness of the proposed method in comparison with existing ones, showing that the proposed method is able to combine satisfactory precision of results with a low computational cost and provide an example to achieve impoverishment prediction and tracking.

On the Theory of a Nonlinear Dynamic Circuit Filtering

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Cointegration between Trends and Their Estimators in State Space Models and Cointegrated Vector Autoregressive Models

A state space model with an unobserved multivariate random walk and a linear observation equation is studied. The purpose is to find out when the extracted trend cointegrates with its estimator, in the sense that a linear combination is asymptotically stationary. It is found that this result holds for the linear combination of the trend that appears in the observation equation. If identifying restrictions are imposed on either the trend or its coefficients in the linear observation equation, it is shown that there is cointegration between the identified trend and its estimator, if and only if the estimators of the coefficients in the observation equations are consistent at a faster rate than the square root of sample size. The same results are found if the observations from the state space model are analysed using a cointegrated vector autoregressive model. The findings are illustrated by a small simulation study.

On Importance Sampling for State Space Models

We consider likelihood inference and state estimation by means of importance sampling for state space models with a nonlinear non-Gaussian observation y ~ p(y|alpha) and a linear Gaussian state alpha ~ p(alpha). The importance density is chosen to be the Laplace approximation of the smoothing density p(alpha|y). We show that computationally efficient state space methods can be used to perform all necessary computations in all situations. It requires new derivations of the Kalman filter and smoother and the simulation smoother which do not rely on a linear Gaussian observation equation. Furthermore, results are presented that lead to a more effective implementation of importance sampling for state space models. An illustration is given for the stochastic volatility model with leverage.

Probabilistic damage detection of structures with uncertainties under unknown excitations based on Parametric Kalman filter with unknown Input

System identification and damage detection for structural health monitoring have received considerable attention. Various time domain analysis methodologies based on measured vibration data of structures have been proposed. Among them, recursive least-squares estimation of structural parameters which is also known as parametric Kalman filter (PKF) approach has been studied. However, the conventional PKF requires that all the external excitations (inputs) be available. On the other hand, structural uncertainties are inevitable for civil infrastructures, it is necessary to develop approaches for probabilistic damage detection of structures. In this paper, a parametric Kalman filter with unknown inputs (PKF-UI) is proposed for the simultaneous identification of structural parameters and the unmeasured external inputs. Analytical recursive formulations of the proposed PKF-UI are derived based on the conventional PKF. Two scenarios of linear observation equations and nonlinear observation equations are discussed, respectively. Such a straightforward derivation of PKF-UI is not available in the literature. Then, the proposed PKF-UI is utilized for probabilistic damage detection of structures by considering the uncertainties of structural parameters. Structural damage index and the damage probability are derived from the statistical values of the identified structural parameters of intact and damaged structure. Some numerical examples are used to validate the proposed method.

The NLMS Algorithm with Time-Variant Optimum Stepsize Derived from a Bayesian Network Perspective

In this letter, we derive a new stepsize adaptation for the normalized least mean square algorithm (NLMS) by describing the task of linear acoustic echo cancellation from a Bayesian network perspective. Similar to the well-known Kalman filter equations, we model the acoustic wave propagation from the loudspeaker to the microphone by a latent state vector and define a linear observation equation (to model the relation between the state vector and the observation) as well as a linear process equation (to model the temporal progress of the state vector). Based on additional assumptions on the statistics of the random variables in observation and process equation, we apply the expectation-maximization (EM) algorithm to derive an NLMS-like filter adaptation. By exploiting the conditional independence rules for Bayesian networks, we reveal that the resulting EM-NLMS algorithm has a stepsize update equivalent to the optimal-stepsize calculation proposed by Yamamoto and Kitayama in 1982, which has been adopted in many textbooks. As main difference, the instantaneous stepsize value is estimated in the M step of the EM algorithm (instead of being approximated by artificially extending the acoustic echo path). The EM-NLMS algorithm is experimentally verified for synthesized scenarios with both, white noise and male speech as input signal.

Nonlinear Filter Designed Suitable for Orbit Estimation

Satellite autonomous orbit determination is a necessary condition for satellite autonomous survival. Spacecraft borne sensors are used for observing common natural celestial to get a linear observation equation after appropriately conversion. It can signiflcantly reduce the calculation and has a more intuitive physical meaning; taking advantage of the special position of starlight vector when refraction occurs to calculate geocentric distance, increase the observation equation information; system model error introduced by flrst-order recursive approximation of two-body orbital kinetic model is one of the key factors that afiect the performance of fllter, a linear combination of the flrst-order derivative function value take place of higher derivatives function can greatly reduce the linearization error of difierential equations. In the design process of fllter, the method is applied to process the state equation, greatly improves the recursive accuracy of the fllter state equation. Simulation results show that, under the same conditions performance of this fllter based on higher order nonlinear orbit estimator is far superior to the flrst order orbit estimator based on extended Kalman flltering technique.

CMT data inversion using a Bayesian information criterion to estimate seismogenic stress fields

SUMMARY We developed an inversion method to estimate the stress fields related to earthquake generation (seismogenic stress fields) from the centroid moment tensors (CMT) of seismic events by using Akaike’s Bayesian information criterion (ABIC). On the idea that the occurrence of an earthquake releases some part of the seismogenic stress field around its hypocentre, we define the CMT of a seismic event by a weighted volume integral of the true but unknown seismogenic stress field. Representing each component of the seismogenic stress field by the superposition of a finite number of 3-D basis functions (tri-cubic B-splines), we obtain a set of linear observation equations to be solved for the expansion coefficients (model parameters). We introduce prior constraint on the roughness of the seismogenic stress field and combine it with observed data to construct a Bayesian model with hierarchic, highly flexible structure controlled by hyper-parameters. The optimum values of the hyper-parameters are objectively determined form observed data by using ABIC. Given the optimum values of the hyper-parameters, we can obtain the best estimates of model parameters by using a maximum likelihood algorithm. We tested the validity of the inversion method through numerical experiments on two synthetic CMT data sets, assuming the distribution of fault orientations to be aligned with the maximum shear stress plane in one case and to be random in the other case. Then we applied the inversion method to actual CMT data in northeast Japan, and obtained the pattern of the seismogenic stress field consistent with geophysical and geological observations.

Non-gyroscope DR and adaptive information fusion algorithm used in GPS/DR device

In view of the problems existing in GPS, a non-gyroscope DR is introduced. The operating principle and the algorithm of the GPS/DR device are also presented. By operating measured data synthetically, linear observation equations are obtained for the information fusion algorithm. This approach avoids model error due to linearizing nonlinear observation equations in the conventional algorithm, so that the stability of information fusion algorithm is improved and computation expenses are reduced. Field running experiments show that satisfactory accuracy can be obtained by the proposed navigation model and algorithm for the non-gyroscope GPS/DR device.

Three-dimensional magnetization vector inversion of a seamount

Abstract A three-dimensional non-uniform magnetic modeling is proposed to obtain information about a magnetization of a seamount, which was divided into many blocks modeled by layered and rectangular prisms, and parameters were assigned to each block describing magnetic three components. Our data were the magnetic total force on the sea. A set of linear observation equations was formulated in terms of three components of magnetization for each block. The solution was obtained by using the conjugate gradients method because of its fast and accurate advantages of calculation. In this inversion, a common set of three components was defined for several blocks to decrease the number of unknown parameters. A computer program has been tested with artificial data and applied to data of Daiichi Kashima Seamount observed during the first phase of the Kaiko project carried out with the R/V Jean Charcot in 1984. In the real application, the crustal structure was divided into three layers (top depth to 5 km depth, 5–6.5 km depth and 6.5–8 km depth). The result of the inversion shows that the top portion and the submerged western half of this seamount are covered with the low magnetization layers, and in the middle layer (5–6.5 km depth) of eastern half side, declinations, inclinations and intensities are almost northward, 15° and 3–5 A/m, respectively. In the third layer (6.5–8 km depth), the reverse magnetizations are revealed in the southeastern and northern sides of Daiichi Kashima Seamount and around Katori Seamount. These reverse magnetizations may reflect part of the magnetic lineations of the Pacific plate.

State Modeling and Estimation of Controllable Noisy Markov Chains

A filtration algorithm of current states of a discrete sequence, optimal by the criterion of the minimum root-mean-square estimation error (MRMSEE), based on a model of linear state and observation equations, introduced for the Markov discrete sequence (Markov chain) was synthesized.

Optimal filtering for bilinear systems and its application to terpolymerization process state identification

The paper presents the optimal non linear filter for bilinear state and linear observation equations confused with white Gaussian disturbances. The general scheme for obtaining the optimal filter in case of polynomial state and linear observation equations is announced. The obtained bilinear filter is applied to solution of the state identification problem for the bilinear terpolymerization process and compared to the optimal linear filter available for the linearized model and to the mixed filter designed as a combination of those filters.

On Optimal Filtering for Polynomial System States

The paper presents the optimal nonlinear filter for quadratic state and linear observation equations confused with white Gaussian disturbances. The general scheme for obtaining the optimal filter in case of polynomial state and linear observation equations is announced.

Automatic on-line measurement of ship's attitude by use of a servo-type accelerometer and inclinometers

For an accurate automatic on-line measurement of the ship's attitude the paper develops an intelligent sensing system which uses one servo-type accelerometer and two servo-type inclinometers which are appropriately located on the ship. By considering the dynamics of the servo-controlled rigid pendulums of the inclinometers, linear observation equations are derived on the rolling and pitching signals of the ship. Moreover, one accelerometer is utilized to extract the heaving signal. Through the introduction of linear dynamic models and the linear observation equations for the three signals, their online measurement is reduced to the state estimation of the linear dynamic systems. A bank of Kalman filters is used to execute the on-line state estimation and to overcome changes in parameters in the dynamic models with time lapse.

On-line sensing system of dynamic ship’s attitude by use of servo-type accelerometers

For an accurate automatic measurement of ship’s attitude the paper proposes an intelligent on-line sensing system which uses four servo-type accelerometers and one servo-type inclinometer appropriately located on the ship. Through an adequate location of the accelerometers, the heaving, rolling, and pitching signals of the ship are separated from each other with adequate linear combinations of the four sensors’ outputs. Furthermore, the inclinometer is utilized to extract a bias signal of the pitching. By introducing linear dynamic models and linear observation equations on the three signals, their on-line measurement is reduced to the state estimation of the linear dynamic systems. A bank of Kalman filters are used to execute the on-line state estimation and to overcome changes in parameters in the dynamic models with time.

Automatic on‐line measurement of ship's attitude by use of servo‐type inclinometers

AbstractAn accurate measurement of dynamic ship's attitude is important for not only correcting data obtained with sonars in searching seabed patterns but also controlling the attitude of a high‐quality ship. From this viewpoint, a previously developed automatic measurement system measured the heaving, rolling, and pitching of a ship by processing outputs of four servotype accelerome‐ters appropriately located on the ship. However, this measurement system required excessive computational time for data processing because it adopted both the maximum entropy method and an optimization procedure to seek the unknown heaving, rolling, and pitching parameters. To realize the on‐line measurement of the three signals, this paper proposes a new automatic measurement system using two servotype inclinometers and one accelerometer. This measurement system introduces linear dynamic models and linear observation equations on the three signals and reduces the on‐line measurement of the three signals to the state estimation of the linear dynamic systems. A bank of Kalman filters is used to execute the on‐line state estimation and to overcome changes in parameters in the dynamic models with time. Since the proposed measurement system utilizes the two servotype inclinometers and one servotype accelerometer, the measurement system will cost less than the previously proposed one.

Learning algorithms for neural networks with the Kalman filters

Based on various approaches, several different learing algorithms have been given in the literature for neural networks. Almost all algorithms have constant learning rates or constant accelerative parameters, though they have been shown to be effective for some practical applications. The learning procedure of neural networks can be regarded as a problem of estimating (or identifying) constant parameters (i.e. connection weights of network) with a nonlinear or linear observation equation. Making use of the Kalman filtering, we derive a new back-propagation algorithm whose learning rate is computed by a time-varying Riccati difference equation. Perceptron-like and correlational learning algorithms are also obtained as special cases. Furthermore, a self-organising algorithm of feature maps is constructed within a similar framework.

Optimal control for a class of partially observable systems†

A continuous time stochastic system with linear dynamics and linear observation equation has to be steered in such a way that the current predicted miss distance of the state to a given hyperplane, evaluated by some cost functional over a finite time interval, is minimized. it is shown that in the class of all controls taking values in the unit cube and depending only on the past of the observation process, the optimal control is bang-bang and that the separation and certainty equivalence principles hold