LM²MSE State Estimation - Kalman Filtering under Stochastic and Unknown but Bounded Uncertainties
Measurement Systems
Final Report Abstract
The project LM²MSE state estimation addresses the question of how to combine different approaches to model uncertainties into a unified concept that can easily be integrated into a Kalman filtering scheme. The proposed solution to this question represents a combination of standard Kalman filtering and ellipsoidal state estimation. In this hybrid filter design, random errors are characterized by mean and covariance matrix while ellipsoidal sets are employed to model unknown but bounded errors. The latter set-membership representation can be exploited by users to model ignorance about error terms. For this purpose, at least boundedness has to be assumed. The combination of stochastic and setmembership error characterizations offers a flexible yet comprehensive way to take into account different sources of estimation uncertainties: While in the original formulation of the Kalman filter the processes are driven by zero-mean Gaussian white noise, additional sensor biases, linearization and quantization errors, or unknown correlations between errors can now easily be considered within the generalized formulation. Even, ellipsoidal constraints on the state space and implicit information can easily be incorporated. Although the proposed estimator, by design, can rather be perceived as an extension of the standard Kalman filter, it includes purely ellipsoidal state estimation as a special case. As a consequence, the LM²MSE filter design generalizes both Kalman and ellipsoidal filtering. This property is supported by a new derivation of the filter as a generalized min-max mean squared error estimator, which is one of the important results of this project. Further project results to be highlighted particularly pertain to applications of the proposed filter, but there is also a surprising insight originating from the fundamental idea behind the project. The proposed filter offers the possibility to incorporate ellipsoidal constraints and implicit information. For example, in the former case, ellipsoidal sets can be exploited to model the constraint that the position of a mobile platform is confined to a certain area. The latter case takes this idea a step further, and missing sensor readings are translated into implicit measurement information. A particular example is event-based state estimation, where a remote sensor triggers transmissions to the estimator when measurements exceed a certain threshold. Missing transmissions can still be exploited in the LM²MSE filter by defining the corresponding ellipsoidal set of possible observations that are below the given threshold. The derived event-based filter shows promising results in a state feedback control design with respect to stability and effectiveness. The surprising insight mentioned above is a new approach to bound unknown common information that is shared by estimates. It turned out that common information is bounded by the intersection of the ellipsoids that are related to the inverse error covariance matrices of the estimates. Based on this insight, a novel fusion algorithm for decentralized state estimation—inverse covariance intersection —has been derived, which is consistent and, at the same time, less conservative than common approaches like covariance intersection. The project has offered several opportunities for valuable and successful collaborations with international researchers. As a result, the derived filter to combine stochastic and set-membership uncertainties has been applied successfully to estimation problems in the field of simultaneous localization and mapping as well as to networked data fusion and control. Also, new research questions, like applications in machine learning, have arisen and will be part of future research.
Publications
- “State Estimation for Ellipsoidally Constrained Dynamic Systems with Set-membership Pseudo Measurements,” in Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2015), San Diego, California, USA, Sep. 2015
B. Noack, M. Baum, and U. D. Hanebeck
(See online at https://doi.org/10.1109/MFI.2015.7295824) - “Treatment of Biased and Dependent Sensor Data in Graph-based SLAM,” in Proceedings of the 18th International Conference on Information Fusion (Fusion 2015), Washington D. C., USA, Jul. 2015
B. Noack, S. J. Julier, and U. D. Hanebeck
- “State Estimation Considering Negative Information with Switching Kalman and Ellipsoidal Filtering,” in Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany, Jul. 2016
B. Noack, F. Pfaff, M. Baum, and U. D. Hanebeck
- “Decentralized Data Fusion with Inverse Covariance Intersection,” Automatica, vol. 79, pp. 35–41, May 2017
B. Noack, J. Sijs, M. Reinhardt, and U. D. Hanebeck
(See online at https://doi.org/10.1016/j.automatica.2017.01.019) - “Event-based State Estimation in a Feedback Loop with Imperfect Communication Links,” in Proceedings of the 20th IFAC World Congress (IFAC 2017), Toulouse, France, Jul. 2017
J. Sijs and B. Noack
(See online at https://doi.org/10.1016/j.ifacol.2017.08.104) - “Optimally Distributed Kalman Filtering with Data-Driven Communication,” Sensors, vol. 18, no. 4, Apr. 2018
K. Dormann, B. Noack, and U. D. Hanebeck
