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In this paper, we propose an approach to address the problems with ambiguity in tuning the process and observation noises for a discrete-time linear Kalman filter. Conventional approaches to tuning (e.g. using normalized estimation error squared and covariance minimization) compute empirical measures of filter performance and the parameter are selected manually or selected using some kind of optimization algorithm to maximize these measures of performance. However, there are two challenges with this approach. First, in theory, many of these measures do not guarantee a unique solution due to observability issues. Second, in practice, empirically computed statistical quantities can be very noisy due to a finite number of samples. We propose a method to overcome these limitations. Our method has two main parts to it. The first is to ensure that the tuning problem has a single unique solution. We achieve this by simultaneously tuning the filter over multiple different prediction intervals. Although this yields a unique solution, practical issues (such as sampling noise) mean that it cannot be directly applied. Therefore, we use Bayesian Optimization. This technique handles noisy data and the local minima that it introduces.
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Many state estimation algorithms must be tuned given the state space process and observation models, the process and observation noise parameters must be chosen. Conventional tuning approaches rely on heuristic hand-tuning or gradient-based optimization techniques to minimize a performance cost function. However, the relationship between tuned noise values and estimator performance is highly nonlinear and stochastic. Therefore, the tuning solutions can easily get trapped in local minima, which can lead to poor choices of noise parameters and suboptimal estimator performance. This paper describes how Bayesian Optimization (BO) can overcome these issues. BO poses optimization as a Bayesian search problem for a stochastic ``black box cost function, where the goal is to search the solution space to maximize the probability of improving the current best solution. As such, BO offers a principled approach to optimization-based estimator tuning in the presence of local minima and performance stochasticity. While extended Kalman filters (EKFs) are the main focus of this work, BO can be similarly used to tune other related state space filters. The method presented here uses performance metrics derived from normalized innovation squared (NIS) filter residuals obtained via sensor data, which renders knowledge of ground-truth states unnecessary. The robustness, accuracy, and reliability of BO-based tuning is illustrated on practical nonlinear state estimation problems,losed-loop aero-robotic control.
This paper studies the distributed state estimation in sensor network, where $m$ sensors are deployed to infer the $n$-dimensional state of a linear time-invariant (LTI) Gaussian system. By a lossless decomposition of optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement and fuses the local estimate of each node with a consensus algorithm. We show that the average of the estimate from all sensors coincides with the optimal Kalman estimate. Numerical examples are provided in the end to illustrate the performance of the proposed scheme.
In this paper, the spacecraft attitude estimation problem has been investigated making use of the concept of matrix Lie group. Through formulation of the attitude and gyroscope bias as elements of SE(3), the corresponding extended Kalman filter, termed as SE(3)-EKF, has been derived. It is shown that the resulting SE(3)-EKF is just the newly-derived geometric extended Kalman filter (GEKF) for spacecraft attitude estimation. This provides a new perspective on the GEKF besides the common frame errors definition. Moreover, the SE(3)-EKF with reference frame attitude error is also derived and the resulting algorithm bears much resemblance to the right invariant EKF.
Kalman filtering has been traditionally applied in three application areas of estimation, state estimation, parameter estimation (a.k.a. model updating), and dual estimation. However, Kalman filter is often not sufficient when experimenting with highly uncertain nonlinear dynamic systems. In this study, a nonlinear estimator is developed by adopting a particle filter algorithm that takes advantage of measured signals. This approach is shown to significantly improve the ability to estimate states. To illustrate this approach, a model for a nonlinear device coupled with a hydraulic actuator plays the role of an actual plant and a nonlinear algebraic function is considered as an approximation of the nonlinear device, thus generating non-parametric and parametric uncertainties. Then we use displacement and force signals to improve the distribution of the states by resampling the set of particles. Finally, all of the states are estimated from these posterior density functions. A set of simulations considering three different noise levels demonstrates that the performance of the particle filter approach is superior to the Kalman filter, yielding substantially better performance when estimating nonlinear physical systems in the presence of modeling uncertainties.
Designing resilient control strategies for mitigating stealthy attacks is a crucial task in emerging cyber-physical systems. In the design of anomaly detectors, it is common to assume Gaussian noise models to maintain tractability; however, this assumption can lead the actual false alarm rate to be significantly higher than expected. We propose a distributionally robust anomaly detector for noise distributions in moment-based ambiguity sets. We design a detection threshold that guarantees that the actual false alarm rate is upper bounded by the desired one by using generalized Chebyshev inequalities. Furthermore, we highlight an important trade-off between the worst-case false alarm rate and the potential impact of a stealthy attacker by efficiently computing an outer ellipsoidal bound for the attack-reachable states corresponding to the distributionally robust detector threshold. We illustrate this trade-off with a numerical example and compare the proposed approach with a traditional chi-squared detector.
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