Filtering Error Covariance Research Articles

Explicit solutions to the singular discrete finite-time linear estimation problem

The problem of optimal filtering of linear discrete-time processes is considered. It is assumed that the driving and measurement noise signals are stationary white, and that the number of measurements is finite. Explicit expressions for the filtering error covariance and the corresponding Kalman gain matrices are obtained for the limiting case where the measurement noise intensity tends uniformly to zero. Similar expressions are derived for the limiting singular smoothing problem with finite number of measurements.

Performance of the Kalman filter with Poisson measurements

In this paper the properties of a Kalman filter, when the measurements arrive at random points in time, are examined under the assumption that the number of measurements arriving in a given time interval can be modelled as a Poisson counting process. The filter error covariance is a function of the times at which the measurements arrive. Since the measurements arrive at random points in time, then the error covariance is a random process. A measure of the filter's performance is the expected value of the error covariance, which is found not to be self-propagating. The exact value of this measure is not found, but computationally feasible bounds are determined to nest it. Using a first-order example it is shown that in the steady-state the bounds are close over all values of the parameter mixes.

Decoupled Kalman filters for phased array radar tracking

Kalman filters have been used in numerous phased array radars to track satellites, reentry vehicles, and missiles. This paper considers the design of these filters to reduce computational requirements, ill-conditioning, and the effects of nonlinearities. Several special coordinate systems used to represent the Kalman filter error covariance matrix are described. These covariance coordinates facilitate the approximate decoupling required for practical filter design. A tutorial discussion and analysis of ill-conditioning in Kalman filters is used to motivate these design considerations. This analysis also explains several well-known phenomena reported in the literature. In addition, a discussion of nonlinearities and methods to mitigate their ill effects is included.

Existence and comparison theorems for partial differential equations of Riccati type

In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.

Measurement updating using the U- D factorization

The discrete linear filtering problem is treated by factoring the filter error covariance matrix as P = UDU T. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As is the case with the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.

Sensitivity analysis of discrete Kalman filters

This paper investigates the sensitivity of discrete Kalman filters to erroneous models. Both parameter and structure (state dimensionality) sensitivity are considered, as well as deterministic and random parameter errors. Iterative algorithms are derived for the calculation of the actual filter error covariance matrix for the case of known (deterministic) modelling errors. For the case of random statistical and dynamical modelling errors, an optimal mean-square error estimate of the actual system performance is derived.