13,240 research outputs found
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering
Recently, the Frank-Wolfe optimization algorithm was suggested as a procedure
to obtain adaptive quadrature rules for integrals of functions in a reproducing
kernel Hilbert space (RKHS) with a potentially faster rate of convergence than
Monte Carlo integration (and "kernel herding" was shown to be a special case of
this procedure). In this paper, we propose to replace the random sampling step
in a particle filter by Frank-Wolfe optimization. By optimizing the position of
the particles, we can obtain better accuracy than random or quasi-Monte Carlo
sampling. In applications where the evaluation of the emission probabilities is
expensive (such as in robot localization), the additional computational cost to
generate the particles through optimization can be justified. Experiments on
standard synthetic examples as well as on a robot localization task indicate
indeed an improvement of accuracy over random and quasi-Monte Carlo sampling.Comment: in 18th International Conference on Artificial Intelligence and
Statistics (AISTATS), May 2015, San Diego, United States. 38, JMLR Workshop
and Conference Proceeding
Robust H∞ filtering for markovian jump systems with randomly occurring nonlinearities and sensor saturation: The finite-horizon case
This article is posted with the permission of IEEE - Copyright @ 2011 IEEEThis paper addresses the robust H∞ filtering problem for a class of discrete time-varying Markovian jump systems with randomly occurring nonlinearities and sensor saturation. Two kinds of transition probability matrices for the Markovian process are considered, namely, the one with polytopic uncertainties and the one with partially unknown entries. The nonlinear disturbances are assumed to occur randomly according to stochastic variables satisfying the Bernoulli distributions. The main purpose of this paper is to design a robust filter, over a given finite-horizon, such that the H∞ disturbance attenuation level is guaranteed for the time-varying Markovian jump systems in the presence of both the randomly occurring nonlinearities and the sensor saturation. Sufficient conditions are established for the existence of the desired filter satisfying the H∞ performance constraint in terms of a set of recursive linear matrix inequalities. Simulation results demonstrate the effectiveness of the developed filter design scheme.This work was supported in part by the National Natural Science Foundation
of China under Grants 61028008, 60825303, and 61004067, National 973 Project under Grant 2009CB320600, the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University) from the Ministry of Education of China, the Engineering and Physical Sciences Research Council (EPSRC) of the U.K., under Grant GR/S27658/01, the Royal Society of the
U.K., and the Alexander von Humboldt Foundation of Germany
Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach
This paper uses stochastic dominance principles to construct upper and lower
sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using
convex optimization methods for nuclear norm minimization with copositive
constraints, we construct low rank stochastic marices so that the optimal
filters using these matrices provably lower and upper bound (with respect to a
partially ordered set) the true filtered distribution at each time instant.
Since these matrices are low rank (say R), the computational cost of evaluating
the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance
sampling filter is presented that exploits these upper and lower bounds to
estimate the optimal posterior. Finally, using the Dobrushin coefficient,
explicit bounds are given on the variational norm between the true posterior
and the upper and lower bounds
Truncated Moment Problem for Dirac Mixture Densities with Entropy Regularization
We assume that a finite set of moments of a random vector is given. Its
underlying density is unknown. An algorithm is proposed for efficiently
calculating Dirac mixture densities maintaining these moments while providing a
homogeneous coverage of the state space.Comment: 18 pages, 6 figure
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