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Register a new userThe universality of the Kalman filter: a conditional characteristic function perspective
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The universality of the celebrated Kalman filtering can be found in control theory. The Kalman filter has found its striking applications in sophisticated autonomous systems and smart products, which are attributed to its realization in a single complex chip. In this paper, we revisit the Kalman filter from the perspective of conditional characteristic function evolution and Ito calculus and develop three Kalman filtering Theorems and their formal proof. Most notably, this paper reveals the following: (i) Kalman filtering equations are a consequence of the evolution of conditional characteristic function for the linear stochastic differential system coupled with the linear discrete measurement system. (ii) The Kalman filtering is a consequence of the stochastic evolution of conditional characteristic function for the linear stochastic differential system coupled with the linear continuous measurement system. (iii) The structure of the Kalman filter remains invariant under two popular stochastic interpretations, the Ito vs Stratonovich.
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Filtering is a data assimilation technique that performs the sequential inference of dynamical systems states from noisy observations. Herein, we propose a machine learning-based ensemble conditional mean filter (ML-EnCMF) for tracking possibly high-dimensional non-Gaussian state models with nonlinear dynamics based on sparse observations. The proposed filtering method is developed based on the conditional expectation and numerically implemented using machine learning (ML) techniques combined with the ensemble method. The contribution of this work is twofold. First, we demonstrate that the ensembles assimilated using the ensemble conditional mean filter (EnCMF) provide an unbiased estimator of the Bayesian posterior mean, and their variance matches the expected conditional variance. Second, we implement the EnCMF using artificial neural networks, which have a significant advantage in representing nonlinear functions over high-dimensional domains such as the conditional mean. Finally, we demonstrate the effectiveness of the ML-EnCMF for tracking the states of Lorenz-63 and Lorenz-96 systems under the chaotic regime. Numerical results show that the ML-EnCMF outperforms the ensemble Kalman filter.
We formulate a recursive estimation problem for multiple dynamical systems coupled through a low dimensional stochastic input, and we propose an efficient sub-optimal solution. The suggested approach is an approximation of the Kalman filter that discards the off diagonal entries of the correlation matrix in its update step. The time complexity associated with propagating this approximate block-diagonal covariance is linear in the number of systems, compared to the cubic complexity of the full Kalman filter. The stability of the proposed block-diagonal filter and its behavior for a large number of systems are analyzed in some simple cases. It is then examined in the context of electric field estimation in a high-contrast space coronagraph, for which it was designed. The numerical simulations provide encouraging results for the cost-efficiency of the newly suggested filter.
Many state estimation and control algorithms require knowledge of how probability distributions propagate through dynamical systems. However, despite hybrid dynamical systems becoming increasingly important in many fields, there has been little work on utilizing the knowledge of how probability distributions map through hybrid transitions. Here, we make use of a propagation law that employs the saltation matrix (a first-order update to the sensitivity equation) to create the Salted Kalman Filter (SKF), a natural extension of the Kalman Filter and Extended Kalman Filter to hybrid dynamical systems. Away from hybrid events, the SKF is a standard Kalman filter. When a hybrid event occurs, the saltation matrix plays an analogous role as that of the system dynamics, subsequently inducing a discrete modification to both the prediction and update steps. The SKF outperforms a naive variational update - the Jacobian of the reset map - by having a reduced mean squared error in state estimation, especially immediately after a hybrid transition event. Compared a hybrid particle filter, the particle filter outperforms the SKF in mean squared error only when a large number of particles are used, likely due to a more accurate accounting of the split distribution near a hybrid transition.
The aim of this paper is to propose a new numerical approximation of the Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is based on the selection of typical trajectories of the driving semi-Markov chain of the process by using an optimal quantization technique. The main advantage of this approach is that it makes pre-computations possible. We derive a Lipschitz property for the solution of the Riccati equation and a general result on the convergence of perturbed solutions of semi-Markov switching Riccati equations when the perturbation comes from the driving semi-Markov chain. Based on these results, we prove the convergence of our approximation scheme in a general infinite countable state space framework and derive an error bound in terms of the quantization error and time discretization step. We employ the proposed filter in a magnetic levitation example with markovian failures and compare its performance with both the Kalman-Bucy filter and the Markovian linear minimum mean squares estimator.
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.
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