A robust Kalman filter is proposed for the discrete-time system with norm-bounded parametric uncertainties. The uncertainties are described by the energy bound constraint, i.e. the sum quadratic constraint (SQC). It is shown that the SQC can be converted into an indefinite quadratic cost function to be minimised in the Krein space, and it is found that the Krein space Kalman filter is a solution of the minimisation problem. After introducing a Krein space state-space model, which includes the uncertainty, one can easily write a robust version of the Krein space Kalman filter by modifying the measurement matrix and the variance of measurement noises in the original Krein space Kalman filter. Since the resulting robust Kalman filter has the same recursive structure as a conventional Kalman filter, a robust filtering scheme can be readily designed using the proposed method. A numerical example demonstrates that the proposed filter achieves robustness against parameter variation and improvement in performance when compared with a conventional Kalman filter and an existing robust Kalman filter, respectively.

Inner Product Structures

Robust Kalman filtering via Krein space estimation

We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal circumradius of spheres. It turns out that a normed vector space is an inner product space if and only if all spheres are not degenerate, i.e., the maximal circumradius of points on the sphere equals the radius of the sphere.

/pdf/a-characterisation-of-inner-product-spaces-by-the-maximal-85u3emzkk6.pdf

A characterisation of inner product spaces by the maximal circumradius of spheres

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

/pdf/characterizations-of-metric-projections-in-banach-spaces-and-5bulkkb9gi.pdf

Characterizations of metric projections in Banach spaces and applications

/pdf/a-brief-walk-through-sampling-theory-2trslmern1.pdf

A Brief Walk Through Sampling Theory

In this paper we introduce two mappings associated with the lower and upper semi- inner product ( ; )i and ( ; )s and with semi-inner products ( ; ) (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and bound- edness of these mappings. We give a renement of the Schwarz inequality, applications to the Birkho orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants. 1 Introduction and preliminaries In this paper we continue the study of mappings associated with the lower and upper semi-inner product in a normed space that started in (DK-97a), (DK-97b) and (DK-98). The main reason for studying these mappings is to obtain sharper estimates than those available using only the norm, and to attempt a clearer geometrical description of the normed space in the absence of a true inner product. We also consider semi-inner products in the sense of Lumer, which lie between the lower and upper semi-inner product, and dene our mappings (; ) x;y in dependence

/pdf/two-mappings-related-to-semi-inner-products-and-their-40iv6jlnl1.pdf

Inner Product Structures

Citations

Robust Kalman filtering via Krein space estimation

A characterisation of inner product spaces by the maximal circumradius of spheres

Characterizations of metric projections in Banach spaces and applications

A Brief Walk Through Sampling Theory

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

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