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The Kalman filter is one of the best-known and most frequently used methods for dynamic state estimation. In addition to a measurement and state transition model, the Kalman filter requires knowledge about the covariance of the measurement and process noise. However, the noise covariances are mostly unknown and may vary during the application. Adaptive Kalman filters solve this problem by estimating the noise covariances online to improve the state estimation. Existing methods are often limited in their application because they are designed to adapt only the measurement noise or the process noise covariance and tend to diverge when both are unknown. Moreover, most methods provide no or only local convergence results, which implies that a poor initialization can adversely affect the estimation of the noise covariances, leading to a deteriorated state estimation. This paper introduces a novel adaptive Kalman filter based on additional Kalman smoothing and analytically derived covariance estimators. Firstly, the unbiased measurement and process noise covariance estimators are derived from the maximum a posteriori formulation of the Kalman smoother. Then, based on these estimators, which depend on the system formulation and the state estimates of the Kalman smoother, the adaptive Kalman filter algorithm is presented. The convergence of the derived estimators can be shown for time-invariant systems for one-dimensional measurement and process noise. For higher-dimensional problems, the convergence can be tested simulatively for the specific dynamical system. A detailed evaluation of various simulation scenarios is presented, demonstrating the accuracy and robustness of the proposed method.

Summary: Structural performance assessment is a critical stage in the health monitoring for the maintenance and risk management of engineering structures, in which the interstory drift is a key indicator. In this paper, a robust adaptive Kalman filter is proposed for an interstory drift estimation problem to show the health condition of steel structures in the case that the statistics or internal dynamics describing the signals and measurements are not known precisely. More precisely, we build an adaptive current Jerk model (ACJM) where the model parameters are updated in each time step to presuppose the statistics characterization of the steel dynamic, while the unknown measurement noise covariance is adapted based on a fixed‐lag innovation with respect to measurements. Moreover, a robust adaptive Kalman filter is designed for the modelling uncertainties in each time increment by solving a minimax game: one player tries to select an "actual" model far from the proposed ACJM with an exponential decay tolerance, while its "hostile" player, namely, the optimum filter, is designed by minimizing the estimation error according to the selected "actual" model. Finally, some simulation and experimental results show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of applications. Although this problem has a long history, reliable algorithms for their estimation were scant, and necessary and sufficient conditions for identifiability of the covariances were in dispute until recently. Necessary and sufficient conditions for covariance estimation and a batch estimation algorithm were presented in our previous study. This paper presents stochastic gradient descent algorithms for noise covariance estimation in adaptive Kalman filters that are an order of magnitude faster than the batch method for similar or better root mean square error. More significantly, these algorithms are applicable to non-stationary systems where the noise covariances can occasionally jump up or down by an unknown magnitude. The computational efficiency of the new algorithms stems from adaptive thresholds for convergence, recursive fading memory estimation of the sample cross-correlations of the innovations, and accelerated stochastic gradient descent algorithms. The comparative evaluation of the proposed methods on a number of test cases demonstrates their computational efficiency and accuracy.

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.

In this letter, we analyze certain student's $t$ -filters for linear Gaussian systems with misspecified noise covariances. It is shown that under appropriate conditions, the filter both estimates the state and re-scales the noise covariance matrices in a Kullback–Leibler optimal fashion. If the noise covariances are misscaled by a common scalar, then the re-scaling is asymptotically exact. We also compare the student's $t$ -filter scale estimates to the maximum-likelihood estimates. Simulations demonstrating the results on the Wiener velocity model are provided. [ABSTRACT FROM AUTHOR]

We revisit the precision of the measurement of track parameters (position, angle) with optimal methods in the presence of detector resolution, multiple scattering and zero magnetic field. We then obtain an optimal estimator of the track momentum by a Bayesian analysis of the filtering innovations of a series of Kalman filters applied to the track. This work could pave the way to the development of autonomous high-performance gas time-projection chambers (TPC) or silicon wafer γ -ray space telescopes and be a powerful guide in the optimization of the design of the multi-kilo-ton liquid argon TPCs that are under development for neutrino studies. [ABSTRACT FROM AUTHOR]

Kalman-based state estimators assumea prioriknowledge of the covariance matrices of the process and observation noise. However, in most practical situations, noise statistics and initial conditions are often unknown and need to be estimated from measurement data. This paper presents an auto-covariance least-squares-based algorithm for noise and initial state error covariance estimation of large-scale linear time-varying (LTV) and nonlinear systems. Compared to existing auto-covariance least-squares based-algorithms, our method does not involve any approximations for LTV systems, has fewer parameters to determine and is more memory/computationally efficient for large-scale systems. For nonlinear systems, our algorithm uses full information estimation/moving horizon estimation instead of the extended Kalman filter, so that the stability and accuracy of noise covariance estimation for nonlinear systems can be guaranteed or improved, respectively. [ABSTRACT FROM AUTHOR]

In order to estimate states from a noise-driven state space system, the state estimator requires a priori knowledge of both process and output noise covariances. Unfortunately, noise statistics are usually unknown and have to be determined from output measurements. Current expectation maximization (EM) based algorithms for estimating noise covariances for nonlinear systems assume the number of additive process and output noise signals are the same as the number of states and outputs, respectively. However, in some applications, the number of additive process noises could be less than the number of states. In this paper, a more general nonlinear system is considered by allowing the number of process and output noises to be smaller or equal to the number of states and outputs, respectively. In order to estimate noise covariances, a semi-definite programming solver is applied, since an analytical solution is no longer easy to obtain. The expectation step in current EM algorithms rely on state estimates from the extended Kalman filter (EKF) or smoother. However, the instability and divergence problems of the EKF could cause the EM algorithm to converge to a local optimum that is far away from true values. We use moving horizon estimation instead of the EKF/smoother so that the accuracy of the covariance estimation in nonlinear systems can be significantly improved. [ABSTRACT FROM AUTHOR]

Autocovariance least-squares (ALS) is a correlation-based noise covariance estimation method that has received much attention recently. However, most existing works focus on the situation without knowledge of the process and measurement noise shaping matrices, i.e., the latter two are taken to be identity matrices in most existing methods. In practice, one might have some prior structural information about how the process/measurement noises affect the system dynamics/measurements. For the above case where the noise shaping matrices are not identity matrices, concise conditions under which the system noise covariances can be uniquely identified have not been discovered so far. To fill the above gap, in this paper, we propose to use deadbeat filters in the ALS framework. By doing so, we will establish concrete sufficient conditions under which one can uniquely identify the process, measurement, and the cross noise covariance matrices. The above results will also be extended to systems with unknown inputs. Albeit the results might be of limited scope, to the best of our knowledge, this is the first time such conditions have been presented in the literature for standard linear systems and systems with unknown inputs. [ABSTRACT FROM AUTHOR]

Pendulum tuned mass dampers (PTMDs) are one of the most popular vibration control devices in use today for flexible towers, bridges, and buildings. Their most attractive feature is the design simplicity; the natural period can be controlled simply by adjusting the suspended length and viscous dampers can be easily integrated into the design. While this simplicity has resulted in their widespread adoption, the installed performance of PTMDs has not been investigated in much detail. This paper presents a methodology for conducting condition assessment of in-service PTMDs. Results from a full-scale study of a PTMD-equipped structure is used as a test bed to demonstrate the approach. The condition of an in-service PTMD is assessed using two criteria: the frequency and damping tuning ratios, and equivalent damping provided by the PTMD. These criteria are estimated while the PTMD is in service, without arresting the PTMD motion, using an extended Kalman filter. Practical considerations typical in full-scale structures arising from model reduction, measurements at limited degrees of freedom, and limited modes of interest are addressed in this assessment framework. [ABSTRACT FROM AUTHOR]

SUMMARY Linear time-invariant systems play significant role in the control field. A number of methods have been published for identification of the deterministic part of a process. However, identification of the stochastic part has had much less attention. This paper deals with estimation of covariance matrices of the noise entering a linear system. The process and measurement noise covariance matrices are tuning parameters of the Kalman filter, and they affect the quality of the state estimation. The noise covariance matrices are generally not known, and their estimation from the measured data is a challenging task. This paper introduces a method for estimation of the noise covariance matrices using Bayesian approach along with Monte Carlo numerical methods. Performance of the approach is tested on various systems and noise properties. The second part of the paper compares Monte Carlo approach with the recently published methods. The speed of convergence is compared with the Cramér-Rao bounds. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In Structural Health Monitoring, Kalman filters can be used as prognosis models, and for damage detection and localization. For a proper functioning, it is necessary to tune these filters with noise covariance matrices for process and measurement noise, which are unknown in practice. Therefore, in the presented work, we apply an autocovariance least-squares method with semidefinite constraints solely based on model parameters. We facilitate this novel approach by formulating the considered innovations covariance function in infinite horizon, which follows inherently from the assumption of linear time-invariant systems. For damage analysis, we adapt a framework based on state-projection estimation errors that was recently established, and yet only applied using H ∞ filters. These estimators represent an alternative to Kalman filters, and are considered robust. Because of this property, the necessity of filter tuning is relaxed, and a naive design is often considered. Based on the damage analysis framework, we derive a new damage indicator that features a high sensitivity towards localized damage. We demonstrate the efficacy of the proposed schemes for noise covariance estimation and damage analysis in a series of simulations inspired by a preceding laboratory test. We finally offer experimental validation, based on vibration test data of a cantilever beam featuring damages at multiple positions, where high sensitivity towards small local stiffness changes is achieved. In our investigations, we compare the damage detection and localization performance of Kalman and H ∞ filters as well as differences in mode shape curvatures (MSC). In the simulation studies, the proposed Kalman filter-based approach outperforms the alternative strategy using H ∞ estimators. The experimental investigations demonstrate a significantly higher sensitivity of the filters towards localized damage compared to differences in MSCs. Considering the totality of investigations, the combined application of both estimators can lead to an increased robustness and sensitivity regarding damage detection and localization. • Parametric noise covariance estimation for Kalman filter-based damage localization. • Damage-sensitive feature for damage assessment by SP2E. • Simulation studies and experimental validation for small stiffness changes. • Comparison to differences is mode shape curvatures. [ABSTRACT FROM AUTHOR]

Existing works in optimal filtering for linear Gaussian systems with arbitrary unknown inputs assume perfect knowledge of the noise covariances in the filter design. This is impractical and raises the question of whether and under what conditions one can identify the noise covariances of linear Gaussian systems with arbitrary unknown inputs. This paper considers the above identifiability question using the correlation-based autocovariance least-squares (ALS) approach. In particular, for the ALS framework, we prove that (i) the process noise covariance Q and the measurement noise covariance R cannot be uniquely jointly identified; (ii) neither Q nor R is uniquely identifiable, when the other is known. This not only helps us to have a better understanding of the applicability of existing filtering frameworks under unknown inputs (since almost all of them require perfect knowledge of the noise covariances) but also calls for further investigation of alternative and more viable noise covariance methods under unknown inputs. Especially, it remains to be explored whether the noise covariances are uniquely identifiable using other correlation-based methods. We are also interested to use regularization for noise covariance estimation under unknown inputs, and investigate the relevant property guarantees for the covariance estimates. The above topics are the main subjects of our current and future work. [ABSTRACT FROM AUTHOR]

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.