Linearly Reconfigurable Kalman Filtering for a Vector Process

TL;DR

This paper introduces a method for dynamically optimizing observation parameters in a linear Kalman filter to minimize mean-squared error, using semidefinite programming and eigen-decomposition techniques.

Contribution

It formulates and solves the problem of parameter reconfiguration in Kalman filtering to improve estimation accuracy under quadratic constraints.

Findings

Optimal solutions via SDP and least-squares for general cases.

Direct eigen-decomposition solution for scalar observations.

Efficient feasibility testing for Min-Max-MSE case.

Abstract

In this paper, we consider a dynamic linear system in state-space form where the observation equation depends linearly on a set of parameters. We address the problem of how to dynamically calculate these parameters in order to minimize the mean-squared error (MSE) of the state estimate achieved by a Kalman filter. We formulate and solve two kinds of problems under a quadratic constraint on the observation parameters: minimizing the sum MSE (Min-Sum-MSE) or minimizing the maximum MSE (Min-Max-MSE). In each case, the optimization problem is divided into two sub-problems for which optimal solutions can be found: a semidefinite programming (SDP) problem followed by a constrained least-squares minimization. A more direct solution is shown to exist for the special case of a scalar observation; in particular, the Min-Sum-MSE solution can be found directly using a generalized eigendecomposition, and is optimally solved utilizing Rayleigh quotient, and the Min-Max-MSE problem reduces to an SDP feasibility test that can be solved via the bisection method.