Steady-state Kalman Filter Gain Research Articles

On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-Old Problem.

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally unknown. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance R via post-fit residuals, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

Integrated system identification and modal state estimation for control of flexible space structures

A novel approach of integrated system identification and modal state estimation is proposed for control of linear dynamical systems including flexible space structures. There are four steps involved in this approach. First, the relation between a stochastic state space model of a dynamical system and the coefficients of its autoregressive model with exogenous input is derived. Second, an adaptive least-squares transversal predictor is used to estimate the coefficients of the model. Third, a state space model and a steady state Kalman filter gain of the dynamical system are then identified from the coefficients of the model by using the eigensystem realization algorithm. Fourth, a modal state estimator is constructed using the modal parameters of the identified model. On-line implementation of this algorithm can continually improve the modal parameters and the filter gain. It can also gradually update the system model when the system characteristics are slowly changing. A numerical example is used to illustrate the feasibility of the new approach.

State-Space Model and Kalman Filter Gain Identification by a Kalman Filter of a Kalman Filter

This paper describes an algorithm that identifies a state-space model and an associated steady-state Kalman filter gain from noise-corrupted input–output data. The model structure involves two Kalman filters where a second Kalman filter accounts for the error in the estimated residual of the first Kalman filter. Both Kalman filter gains and the system state-space model are identified simultaneously. Knowledge of the noise covariances is not required.

Depth sensor and DVL‐aided vertical channel damping loop design for underwater vehicles

A novel vertical channel damping loop design method that uses not only a depth sensor but also a Doppler velocity log (DVL) for underwater vehicles is proposed. In the proposed method, the loop gains are calculated by adopting the steady-state Kalman filter gain with consideration of characteristics of the depth sensor and the DVL. The simulation results demonstrate that the proposed method achieves better performance than the conventional method.

Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain

The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain.

Iterative LQG Controller Design Through Closed-Loop Identification

This paper presents an iterative LQG controller design approach for a linear stochastic system with an uncertain openloop model and unknown noise statistics. This approach consists of closed-loop identification and controller redesign cycles. In each cycle, the closed-loop identification method is used to identify an open-loop model and a steady-state Kalman filter gain from closed-loop input/output test data obtained by using a feedback LQG controller designed from the previous cycle. Then the identified open-loop model is used to redesign the state feedback. The state feedback and the identified Kalman filter gain are used to form an updated LQG controller for the next cycle. This iterative process continues until the updated controller converges. The proposed controller design is demonstrated by numerical simulations and experiments on a highly unstable large-gap magnetic suspension system.

An alternate calculation of the discrete-time Kalman filter gain and Riccati equation solution

We describe an algorithm to calculate the steady-state Kalman filter gain and Riccati equation solution for a discrete-time Kalman filter. Our algorithm makes use of an approximate autoregressive model for the one-step predictor and only requires the solutions to linear equations. All of the nonlinear calculations can be made explicitly.

Estimation of Steady-State Optimal Filter Gain From Nonoptimal Kalman Filter Residuals

This paper proposes a new algorithm to estimate the optimal steady-state Kalman filter gain of a linear, discrete-time, time-invariant stochastic system from nonoptimal Kalman filter residuals. The system matrices are known, but the covariances of the white process and measurement noises are unknown. The algorithm first derives a moving average (MA) model which relates the optimal and nonoptimal residuals. The MA model is then approximated by inverting a long autoregressive (AR) model. From the MA parameters the Kalman filter gain is calculated. The estimated gain in general is suboptimal due to the approximations involved in the method and a finite number of data. However, the numerical example shows that the estimated gain could be near optimal.

Relationship between steady state Kalman filter gain and noise variances

The Kalman filter requires an exact knowledge of the noise covariance matrices Theoretically, they may take arbitrary values under some restrictions ; positive semidefinite or positive definite. Values of the noise covariance matrices have direct effects upon the Kalman filter gain, and therefore affect the final result, of estimation. Practically, the noise covariance matrices are either unknown or are known only approximately, so they are often determined in a rule of trial and error. In this paper, we discover interesting relations between an index λ and the noise covariance matrices for multivariable identity transition systems and for general linear dynamic systems, through an algorithm of the exponential weighted least squares method. These relations are useful for determining the noise covariance matrices. New results are summarized into three points. The first main result is the relations between the steady state Kalman filter gain and the noise covariance matrices for multivariable identity trans...

Estimation of steady-state Kalman filter gain

An external description of multivariable linear stochastic systems gives a new estimation method of the steady, state Kalman filter gain for systems with unknown noise covariances.