observation equation
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2021 ◽  
Vol 7 (1) ◽  
pp. 160
Author(s):  
Polina A. Yurovskikh

We consider a set membership estimation problem for linear non-stationary systems for which initial states belong to a compact set and uncertain disturbances in an observation equation are integrally restricted. We provethat the exact information set of the system can be approximated by a set of external ellipsoids in the absence of disturbances in the dynamic equation.There are three examples of linear systems. Two examples illustrate the main theorem of the paper, the latter one shows the possibility of generalizing the theorem to the case with disturbances in the dynamic equation.


Geomatics ◽  
2021 ◽  
Vol 1 (3) ◽  
pp. 324-334
Author(s):  
Thomas H. Meyer ◽  
Ahmed F. Elaksher

The process of positioning, using only distances from control stations, is called trilateration (or multilateration if the problem is over-determined). The observation equation is Pythagoras’s formula, in terms of the summed squares of coordinate differences and, thus, is nonlinear. There is one observation equation for each control station, at a minimum, which produces a system of simultaneous equations to solve. Over-determined nonlinear systems of simultaneous equations are typically solved using iterative least squares after forming the system as a truncated Taylor’s series, omitting the nonlinear terms. This paper provides a linearization of the observation equation that is not a truncated infinite series—it is exact—and, thus, is solved exactly, with full rigor, without iteration and, thus, without the need of first providing approximate coordinates to seed the iteration. However, there is a cost of requiring an additional observation beyond that required by the non-linear approach. The examples and terminology come from terrestrial land surveying, but the method is fully general: it works for, say, radio beacon positioning, as well. The approach can use slope distances directly, which avoids the possible errors introduced by atmospheric refraction into the zenith-angle observations needed to provide horizontal distances. The formulas are derived for two- and three-dimensional cases and illustrated with an example using total-station and global navigation satellite system (GNSS) data.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1445
Author(s):  
Rodi Lykou ◽  
George Tsaklidis

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.


2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ming Yan ◽  
Zengcai Wang

The key technology to realize intelligent unmanned coal mining is the strapdown inertial navigation system (SINS); however, the gradual increase of cumulative error during the working process needs to be solved. On the basis of an SINS/odometer (OD)-integrated navigation system, this paper adds magnetometer (MAG)-aided positioning and proposes an SINS/OD/MAG-integrated shearer navigation system. The velocity observation equation is obtained from the speed constraints during shearer movement, and the yaw angle observation equation is obtained from the magnetometer output. The position information of the SINS output is calibrated using these two observations. In order to improve the fault tolerance of the integrated navigation system, an adaptive federated Kalman filter is established to complete the data fusion of the SINS. Experimental results show that the positioning accuracy of the SINS/OD/MAG-integrated navigation system is 75.64% and 74.01% higher in the east and north directions, respectively, than the SINS/OD-integrated navigation system.


2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Zhouming Yang ◽  
Xin Liu ◽  
Jinyun Guo ◽  
Yaowei Xia ◽  
Xiaotao Chang

Cycle slip detection and repair play important roles in the processing of data from dual-frequency GPS receivers onboard low-Earth orbit (LEO) satellites. To detect and repair cycle slips more comprehensively, an enhanced error method (EEM) is proposed. EEM combines single-frequency and narrow-lane carrier phase observations to construct special observations and observation equation groups. These special observations differ across time and satellite (ATS). ATS observations are constructed by three steps. The first step is differencing single-frequency and narrow-lane observations through a time difference (TD). The second step is to select a satellite as a reference satellite and other satellites as nonreference satellites. The third step is to difference the single-frequency TD observations from the reference satellite and the narrow-lane TD observations from the nonreference satellites by a satellite difference. If cycle slips occur at the reference satellite, the correction values for these ATS observations can be significantly enlarged. To process all satellites, the EEM selects each satellite as a reference satellite and builds the corresponding equation group. The EEM solves these observation equation groups according to the weighted least-squares adjustment (LSA) criterion and obtains the correction values; these correction values are then used to construct the χ 2 values corresponding to different equation groups, and the EEM subsequently carries out a chi-square distribution test for these χ 2 . The satellite corresponding to the maximum χ 2 will be marked. Then, the EEM iteratively processes the other satellites. Cycle slips can be estimated by rounding the float solutions of changes in the ambiguities of cycle slip satellites to the nearest integer. The simulation test results show that the EEM can be used to detect special cycle slip pairs such as (1, 1) and (9, 7). The EEM needs only observation data in two adjacent epochs and is still applicable to observation epochs with continuous cycle slips.


2020 ◽  
Vol 8 ◽  
Author(s):  
Shun-ichi Watanabe ◽  
Tadashi Ishikawa ◽  
Yusuke Yokota ◽  
Yuto Nakamura

Global Navigation Satellite System–Acoustic ranging combined seafloor geodetic technique (GNSS-A) has extended the geodetic observation network into the ocean. The key issue for analyzing the GNSS-A data is how to correct the effect of sound speed variation in the seawater. We constructed a generalized observation equation and developed a method to directly extract the gradient sound speed structure by introducing appropriate statistical properties in the observation equation, especially the data correlation term. In the proposed scheme, we calculate the posterior probability based on the empirical Bayes approach using the Akaike’s Bayesian Information Criterion for model selection. This approach enabled us to suppress the overfitting of sound speed variables and thus to extract simpler sound speed field and stable seafloor positions from the GNSS-A dataset. The proposed procedure is implemented in the Python-based software “GARPOS” (GNSS-Acoustic Ranging combined POsitioning Solver).


2020 ◽  
Author(s):  
tieding lu

<p> Uncertainties usually exist in the process of acquisition of measurement data, which affect the results of the parameter estimation. The solution of the uncertainty adjustment model can effectively improve the validity and reliability of parameter estimation. When the coefficient matrix of the observation equation has a singular value close to zero, i.e., the coefficient matrix is ill-posed, the ridge estimation can effectively suppress the influence of the ill-posed problem of the observation equation on the parameter estimation. When the uncertainty adjustment model is ill-posed, it is more seriously affected by the error of the coefficient matrix and observation vector. In this paper, the ridge estimation method is applied to ill-posed uncertainty adjustment model, deriving an iterative algorithm to improve the stability and reliability of the results. The derived algorithm is verified by two examples, and the results show that the new method is effective and feasible.</p>


2020 ◽  
Vol 19 (03) ◽  
pp. 2050022
Author(s):  
Dhruvi S. Bhatt ◽  
Shaival H. Nagarsheth ◽  
Shambhu N. Sharma

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.


2019 ◽  
Vol 19 (12) ◽  
pp. 1950156 ◽  
Author(s):  
Jia He ◽  
Xiaoxiong Zhang ◽  
Bin Xu

The identification of parameters of linear or nonlinear systems under unknown inputs and limited outputs is an important but still challenging topic in the context of structural health monitoring. Time-domain analysis methodologies, such as extend Kalman filter (EKF), have been actively studied and shown to be powerful for parameter identification. However, the conventional EKF is not applicable when the input is unknown or unmeasured. In this paper, by introducing a projection matrix in the observation equation, a time-domain EKF-based approach is proposed for the simultaneous identification of structural parameters and the unknown excitations with limited outputs. A revised version of observation equation is presented. The unknown inputs are identified using the least squares estimation based on the limited observations and the estimated structural parameters at the current time step. Particularly, an analytical recursive solution is derived. The accuracy and effectiveness of the proposed approach is first demonstrated via several numerical examples. Then it was validated by the shaking table tests on a five-story building model for the robustness in application to real structures. The results show that the proposed approach can satisfactorily identify the parameters of linear or nonlinear structures under unknown inputs.


2019 ◽  
Vol 9 (2) ◽  
pp. 306 ◽  
Author(s):  
Jia He ◽  
Xiaoxiong Zhang ◽  
Mengchen Qi ◽  
Bin Xu

Nonlinearity exists widely in civil engineering structures; for example, the initiation and growth of damage under dynamic loadings is a typical nonlinear process. To date, for the purpose of structural evaluation and a better understanding nonlinear characteristics of complicated structures, a number of parametric and nonparametric methods have been developed for the identification of nonlinear restoring force (NRF). However, due to the highly individualistic nature of nonlinear systems, it would be inefficient to attempt to express the structural NRF in a general parametric form. For many nonparametric techniques, their nonparametric models or approximations may result in undesirable results or oscillations around unsmooth points. In this paper, on the basis of extended Kalman filter (EKF), a model-free NRF identification approach is proposed to circumvent the limitations mentioned above. The NRF to be identified was treated as ‘unknown fictitious input’, and thus, no prior assumptions or approximations for the NRF model were required. With the aid of a projection matrix, a modified version of observation equation was obtained. Based on the principle of EKF, the recursive solution of the proposed approach was analytically derived. The NRFs provided by the nonlinear components were identified by means of least squares estimation (LSE) at each time step. Numerical examples, including building structures equipped with magnetorheological (MR) damper and shape memory alloy (SMA) damper, demonstrated that the proposed approach is capable of satisfactorily identifying NRF without knowledge or intuitive assumptions of any nonlinear model class in advance.


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