3,667 research outputs found
The geometry of low-rank Kalman filters
An important property of the Kalman filter is that the underlying Riccati
flow is a contraction for the natural metric of the cone of symmetric positive
definite matrices. The present paper studies the geometry of a low-rank version
of the Kalman filter. The underlying Riccati flow evolves on the manifold of
fixed rank symmetric positive semidefinite matrices. Contraction properties of
the low-rank flow are studied by means of a suitable metric recently introduced
by the authors.Comment: Final version published in Matrix Information Geometry, pp53-68,
Springer Verlag, 201
A Contraction Analysis of the Convergence of Risk-Sensitive Filters
A contraction analysis of risk-sensitive Riccati equations is proposed. When
the state-space model is reachable and observable, a block-update
implementation of the risk-sensitive filter is used to show that the N-fold
composition of the Riccati map is strictly contractive with respect to the
Riemannian metric of positive definite matrices, when N is larger than the
number of states. The range of values of the risk-sensitivity parameter for
which the map remains contractive can be estimated a priori. It is also found
that a second condition must be imposed on the risk-sensitivity parameter and
on the initial error variance to ensure that the solution of the risk-sensitive
Riccati equation remains positive definite at all times. The two conditions
obtained can be viewed as extending to the multivariable case an earlier
analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure
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
Online identification and nonlinear control of the electrically stimulated quadriceps muscle
A new approach for estimating nonlinear models of the electrically stimulated quadriceps muscle group under nonisometric conditions is investigated. The model can be used for designing controlled neuro-prostheses. In order to identify the muscle dynamics (stimulation pulsewidth-active knee moment relation) from discrete-time angle measurements only, a hybrid model structure is postulated for the shank-quadriceps dynamics. The model consists of a relatively well known time-invariant passive component and an uncertain time-variant active component. Rigid body dynamics, described by the Equation of Motion (EoM), and passive joint properties form the time-invariant part. The actuator, i.e. the electrically stimulated muscle group, represents the uncertain time-varying section. A recursive algorithm is outlined for identifying online the stimulated quadriceps muscle group. The algorithm requires EoM and passive joint characteristics to be known a priori. The muscle dynamics represent the product of a continuous-time nonlinear activation dynamics and a nonlinear static contraction function described by a Normalised Radial Basis Function (NRBF) network which has knee-joint angle and angular velocity as input arguments. An Extended Kalman Filter (EKF) approach is chosen to estimate muscle dynamics parameters and to obtain full state estimates of the shank-quadriceps dynamics simultaneously. The latter is important for implementing state feedback controllers. A nonlinear state feedback controller using the backstepping method is explicitly designed whereas the model was identified a priori using the developed identification procedure
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
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