Introduction: usually there are some outliers (abnormal measurements) in observed data, and they can significantly affect the quality of the data processing. Many dynamic processes are described with stochastic nonlinear equations. Modern nonlinear filters that include the cubature Kalman filter, which deserves a special attention, cannot effectively process data containing abnormal measurements. One of the possible solutions to this problem is to use so-called robust methods that have good performance when one has to analyze data containing outliers. The paper deals with the common situations, when the considered process is actually continuous, but the observed data is taken discretely. Purpose: identifying the most effective advanced robust modifications of the continuous-discrete cubature Kalman filter and giving the appropriate recommendations for their appliance. Results: four modifications of the continuous-discrete cubature Kalman filter have been proposed based on the variational Bayesian and correntropy robust approaches to parameter estimation for stochastic processes. All the modifications have parameters with optimal values depending on both the selected mathematical model and the considered set of observations composing the sample. These values are determined numerically by minimizing the accumulated root mean square error on some grid. The research on the effectiveness of the proposed robust modifications has been carried out for the problem of tracking a space vehicle during its reentry into the atmosphere. The stochastic and the grouped outliers have been considered. Two most effective filters that have approximately equal qualities of estimation have been derived. The correntropy filter that has one configurable parameter can be recommended for practical using.

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Предложены две устойчивые к ошибкам машинного округления и к аномальным данным квадратно-корневые модификации непрерывно-дискретного кубатурного фильтра Калмана, основанные на вариационном байесовском и коррентропийном подходах. Апробация разработанных алгоритмов на модельной задаче со случайным характером расположения аномальных наблюдений показала их работоспособность при сопоставимом качестве фильтрации. Подтверждена алгебраическая эквивалентность представленных квадратно-корневых и стандартных версий Rounding errors due to the finite length of machine word can significantly affect the quality of estimation and filtering when solving the corresponding problems in various subject areas. In this regard, to improve the reliability of the obtained results, it is advisable to develop and then apply square-root modifications of the used algorithms. Purpose: developing the square-root modifications of the continuous-discrete cubature Kalman filter on the basis of variational Bayesian and correntropy approaches. Methodology: matrix orthogonal QR decomposition. Findings: two robust (resistant to the possible presence of anomalous data and to machine rounding errors) modifications of the continuous-discrete cubature Kalman filter have been developed. The first (variational Bayesian) algorithm is obtained by extending the known discrete equations of the extrapolation stage to the continuous-discrete case. The second algorithm, based on the maximum correntropy criterion, is proposed in this paper for the first time. The developed square-root algorithms for nonlinear filtering are validated on the example of one stochastic dynamical system model with the random location of anomalous observations. In doing so, the filtering quality, estimated by the value of the accumulated mean square error, was quite comparable for both modifications during equivalent results obtained for the corresponding root-free analogues. Value: the proposed square-root versions of robust modifications of the continuous-discrete cubature Kalman filter are algebraically equivalent to their standard analogues. Meanwhile, positive definiteness and symmetry of covariance matrices of the state vector estimates at the extrapolation and the filtration stages are provided. The developed algorithms will be used to develop software and mathematical support for parametric identification of stochastic nonlinear continuous-discrete systems in the presence of anomalous observations in the measurement data

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