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Unscented Kalman filters (UKFs) have become popular in the research community. Most UKFs work only with Euclidean systems, but in many scenarios it is advantageous to consider systems with state-variables taking values on Riemannian manifolds. However, we can still find some gaps in the literature's theory of UKFs for Riemannian systems: for instance, the literature has not yet developed first, Riemannian extensions of some fundamental concepts of the UKF theory (e.g., extensions of $\sigma$ -representation, unscented transformation, additive UKF, augmented UKF, additive-noise system), second, proofs of some steps in their UKFs for Riemannian systems (e.g., proof of sigma points parameterization by vectors, state correction equations, noise statistics inclusion), and third, relations between their UKFs for Riemannian systems. In this paper, we attempt to develop a theory capable of filling these gaps. Among other results, we propose Riemannian extensions of the main concepts in the UKF theory (including closed forms), justify all steps of the proposed UKFs, and provide a framework able to relate UKFs for particular manifolds among themselves and with UKFs for Euclidean spaces. Compared with UKFs for Riemannian manifolds of the literature, the proposed filters are more consistent, formally principled, and general. An example of satellite attitude tracking illustrates the proposed theory. [ABSTRACT FROM AUTHOR]

Finsler's lemma is a classic mathematical result with applications in control and optimization. When the lemma is applied to parameter-dependent LMIs, as such those that arise from problems of robust stability, the extra variables introduced by this lemma also become dependent on this parameter. This technical note presents some sufficient conditions which ensure, without losing generality, that these extra variables can assume a simple functional dependence on the parameters as continuity or even independence. The results allow avoiding an unnecessary use of a more functionally complicated parameter-dependent variable that increases the search computational burden without reducing the conservatism of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and cross-covariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-of-the-art methods. [ABSTRACT FROM PUBLISHER]

In this technical note, a robust recursive regulator for linear discrete-time systems, which are subject to parametric uncertainties, is proposed. The main feature of the optimal regulator developed is the absence of tuning parameters in online applications. To achieve this purpose, a quadratic cost function based on the combination of penalty function and robust weighted least-squares methods is formulated. The convergence and stability proofs for the stationary system and a numerical comparative study among the standard linear quadratic regulator, guaranteed cost and H\infty controllers are provided. [ABSTRACT FROM PUBLISHER]