Convergence analysis of a family of robust Kalman filters based on the contraction principle

In this paper we analyze the convergence of a family of robust Kalman filters. For each filter of this family the model uncertainty is tuned according to the so called tolerance parameter. Assuming that the corresponding state-space model is reachable and observable, we show that the corresponding Riccati-like mapping is strictly contractive provided that the tolerance is sufficiently small, accordingly the filter converges

Robust Kalman Filtering: Asymptotic Analysis of the Least Favorable Model

We consider a robust filtering problem where the robust filter is designed according to the least favorable model belonging to a ball about the nominal model. In this approach, the ball radius specifies the modeling error tolerance and the least favorable model is computed by performing a Riccati-like backward recursion. We show that this recursion converges provided that the tolerance is sufficiently small

Model Predictive Control meets robust Kalman filtering

Model Predictive Control (MPC) is the principal control technique used in industrial applications. Although it offers distinguishable qualities that make it ideal for industrial applications, it can be questioned its robustness regarding model uncertainties and external noises. In this paper we propose a robust MPC controller that merges the simplicity in the design of MPC with added robustness. In particular, our control system stems from the idea of adding robustness in the prediction phase of the algorithm through a specific robust Kalman filter recently introduced. Notably, the overall result is an algorithm very similar to classic MPC but that also provides the user with the possibility to tune the robustness of the control. To test the ability of the controller to deal with errors in modeling, we consider a servomechanism system characterized by nonlinear dynamics

Contracting Nonlinear Observers: Convex Optimization and Learning from Data

A new approach to design of nonlinear observers (state estimators) is proposed. The main idea is to (i) construct a convex set of dynamical systems which are contracting observers for a particular system, and (ii) optimize over this set for one which minimizes a bound on state-estimation error on a simulated noisy data set. We construct convex sets of continuous-time and discrete-time observers, as well as contracting sampled-data observers for continuous-time systems. Convex bounds for learning are constructed using Lagrangian relaxation. The utility of the proposed methods are verified using numerical simulation.Comment: conference submissio

Stability of Filters for the Navier-Stokes Equation

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal

On the Robustness of the Bayes and Wiener Estimators under Model Uncertainty

This paper deals with the robust estimation problem of a signal given noisy observations. We assume that the actual statistics of the signal and observations belong to a ball about the nominal statistics. This ball is formed by placing a bound on the Tau-divergence family between the actual and the nominal statistics. Then, the robust estimator is obtained by minimizing the mean square error according to the least favorable statistics in that ball. Therefore, we obtain a divergence family-based minimax approach to robust estimation. We show in the case that the signal and observations have no dynamics, the Bayes estimator is the optimal solution. Moreover, in the dynamic case, the optimal offline estimator is the noncausal Wiener filter

Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation