40,974 research outputs found
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Nonlinear stability and ergodicity of ensemble based Kalman filters
The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are
data assimilation methods used to combine high dimensional, nonlinear dynamical
models with observed data. Despite their widespread usage in climate science
and oil reservoir simulation, very little is known about the long-time behavior
of these methods and why they are effective when applied with modest ensemble
sizes in large dimensional turbulent dynamical systems. By following the basic
principles of energy dissipation and controllability of filters, this paper
establishes a simple, systematic and rigorous framework for the nonlinear
analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the
dynamical properties of boundedness and geometric ergodicity. The time uniform
boundedness guarantees that the filter estimate will not diverge to machine
infinity in finite time, which is a potential threat for EnKF and ESQF known as
the catastrophic filter divergence. Geometric ergodicity ensures in addition
that the filter has a unique invariant measure and that initialization errors
will dissipate exponentially in time. We establish these results by introducing
a natural notion of observable energy dissipation. The time uniform bound is
achieved through a simple Lyapunov function argument, this result applies to
systems with complete observations and strong kinetic energy dissipation, but
also to concrete examples with incomplete observations. With the Lyapunov
function argument established, the geometric ergodicity is obtained by
verifying the controllability of the filter processes; in particular, such
analysis for ESQF relies on a careful multivariate perturbation analysis of the
covariance eigen-structure.Comment: 38 page
State Estimation for a Humanoid Robot
This paper introduces a framework for state estimation on a humanoid robot
platform using only common proprioceptive sensors and knowledge of leg
kinematics. The presented approach extends that detailed in [1] on a quadruped
platform by incorporating the rotational constraints imposed by the humanoid's
flat feet. As in previous work, the proposed Extended Kalman Filter (EKF)
accommodates contact switching and makes no assumptions about gait or terrain,
making it applicable on any humanoid platform for use in any task. The filter
employs a sensor-based prediction model which uses inertial data from an IMU
and corrects for integrated error using a kinematics-based measurement model
which relies on joint encoders and a kinematic model to determine the relative
position and orientation of the feet. A nonlinear observability analysis is
performed on both the original and updated filters and it is concluded that the
new filter significantly simplifies singular cases and improves the
observability characteristics of the system. Results on simulated walking and
squatting datasets demonstrate the performance gain of the flat-foot filter as
well as confirm the results of the presented observability analysis.Comment: IROS 2014 Submission, IEEE/RSJ International Conference on
Intelligent Robots and Systems (2014) 952-95
Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter
We consider the problem of an ensemble Kalman filter when only partial
observations are available. In particular we consider the situation where the
observational space consists of variables which are directly observable with
known observational error, and of variables of which only their climatic
variance and mean are given. To limit the variance of the latter poorly
resolved variables we derive a variance limiting Kalman filter (VLKF) in a
variational setting. We analyze the variance limiting Kalman filter for a
simple linear toy model and determine its range of optimal performance. We
explore the variance limiting Kalman filter in an ensemble transform setting
for the Lorenz-96 system, and show that incorporating the information of the
variance of some un-observable variables can improve the skill and also
increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure
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