In general, theKalman filterrobust optimal filter forRobustKalman filter nonlinear stochastic systems is often too complex to handle. Moreover, for non-Gaussian stochastic environment, the commonly utilized expectation-covariance representation may not suffice. Particularly, in many practical examples, the real noise probability density function (pdf) differs from the supposed Gaussian one by heavier tails. This, in turn, generates the rare spiky noise realizations, named observation outliers. Therefore, it is of great practical interest to design robustified Kalman filters that can cope with outliers. Since the optimal Kalman filter is the linear function of observations, it is susceptible to impulsive noise, or outliers, contaminating the mainly Gaussian distributed observations. In this sense, robust statistical methods provide for suitable tools to spot bad data points and suppress their effects. Statistical literature abounds by different concepts of robustness, and many robust schemes are derived from nonlinear regression problems. Particularly, the Huber’sHuber M-robust statistical approach is widely used in practice, since it represents an approximation of the optimal maximum likelihood (ML) estimationEstimatormaximum likelihood (ML) that is natural and easy to implement. The concept based on using combination of the M-robust estimator with the optimal Kalman filter, or least-squares estimator, has long history, and various robust variants of the optimal Kalman filters are available in the literature. In this sense, the proposed robust estimators may be classified into the two groups. The first one is a family of the nonrecursive, batch processing or off-line robust schemes, where the Kalman filtering problem is recast as the parameter regression one, which is solved by the Huber’s M-robust estimator. The posed optimization problem is nonlinear, and an iterative numerical method is required to solve it. Thus, the standard or simplified Newton’s method, as well the iteratively reweighted least square method are recommended. A such-derived robust estimator is in a batch-mode regression form, processing the observation and the predictions simultaneously, that makes it very effective in suppressing the outliers. However, the robustness in these estimators is achieved at the coast of increasing computational requirements. The computational complexity basically depends on the number of necessary iterations to solve the regression problem. Starting from the computational considerations, the second group represents a family of the parameter and state estimators that calculate an estimate recursively, or sequentially, because of the practical requirements to on-line, or real-time, signal processing. Thus, a recursive robust estimator represents an acceptable balance between the computational efforts and the practical robustness performances. Both the nonrecursive and recursive M-robustified Kalman filtering techniques are considered in the sequel. Finally, the min-max robust approach to designing a recursive robust version of the optimal Kalman filter are also included.

Robust Kalman Filtering

章节
发表日期：

2026年

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Branko Kovačević,
Željko Đurović,
Zoran Banjac

摘要

In general, theKalman filterrobust optimal filter forRobustKalman filter nonlinear stochastic systems is often too complex to handle. Moreover, for non-Gaussian stochastic environment, the commonly utilized expectation-covariance representation may not suffice. Particularly, in many practical examples, the real noise probability density function (pdf) differs from the supposed Gaussian one by heavier tails. This, in turn, generates the rare spiky noise realizations, named observation outliers. Therefore, it is of great practical interest to design robustified Kalman filters that can cope with outliers. Since the optimal Kalman filter is the linear function of observations, it is susceptible to impulsive noise, or outliers, contaminating the mainly Gaussian distributed observations. In this sense, robust statistical methods provide for suitable tools to spot bad data points and suppress their effects. Statistical literature abounds by different concepts of robustness, and many robust schemes are derived from nonlinear regression problems. Particularly, the Huber’sHuber M-robust statistical approach is widely used in practice, since it represents an approximation of the optimal maximum likelihood (ML) estimationEstimatormaximum likelihood (ML) that is natural and easy to implement. The concept based on using combination of the M-robust estimator with the optimal Kalman filter, or least-squares estimator, has long history, and various robust variants of the optimal Kalman filters are available in the literature. In this sense, the proposed robust estimators may be classified into the two groups. The first one is a family of the nonrecursive, batch processing or off-line robust schemes, where the Kalman filtering problem is recast as the parameter regression one, which is solved by the Huber’s M-robust estimator. The posed optimization problem is nonlinear, and an iterative numerical method is required to solve it. Thus, the standard or simplified Newton’s method, as well the iteratively reweighted least square method are recommended. A such-derived robust estimator is in a batch-mode regression form, processing the observation and the predictions simultaneously, that makes it very effective in suppressing the outliers. However, the robustness in these estimators is achieved at the coast of increasing computational requirements. The computational complexity basically depends on the number of necessary iterations to solve the regression problem. Starting from the computational considerations, the second group represents a family of the parameter and state estimators that calculate an estimate recursively, or sequentially, because of the practical requirements to on-line, or real-time, signal processing. Thus, a recursive robust estimator represents an acceptable balance between the computational efforts and the practical robustness performances. Both the nonrecursive and recursive M-robustified Kalman filtering techniques are considered in the sequel. Finally, the min-max robust approach to designing a recursive robust version of the optimal Kalman filter are also included.