3,678 research outputs found
Practical stability with respect to model mismatch of approximate discrete-time output feedback control
This paper establishes a practical stability result for discrete-time output feedback control involving mismatch between the exact system to be stabilised and the approximating system used to design the controller. The practical stability is in the sense of an asymptotic bound on the amount of error bias introduced by the model approximation, and is established using local consistency properties of the systems. Importantly, the practical stability established here does not require the approximating system to be of the same model type as the exact system. Examples are presented to illustrate the nature of our practical stability result
Inverse Problems and Data Assimilation
These notes are designed with the aim of providing a clear and concise
introduction to the subjects of Inverse Problems and Data Assimilation, and
their inter-relations, together with citations to some relevant literature in
this area. The first half of the notes is dedicated to studying the Bayesian
framework for inverse problems. Techniques such as importance sampling and
Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the
desirable property that in the limit of an infinite number of samples they
reproduce the full posterior distribution. Since it is often computationally
intensive to implement these methods, especially in high dimensional problems,
approximate techniques such as approximating the posterior by a Dirac or a
Gaussian distribution are discussed. The second half of the notes cover data
assimilation. This refers to a particular class of inverse problems in which
the unknown parameter is the initial condition of a dynamical system, and in
the stochastic dynamics case the subsequent states of the system, and the data
comprises partial and noisy observations of that (possibly stochastic)
dynamical system. We will also demonstrate that methods developed in data
assimilation may be employed to study generic inverse problems, by introducing
an artificial time to generate a sequence of probability measures interpolating
from the prior to the posterior
Nudging the particle filter
We investigate a new sampling scheme aimed at improving the performance of
particle filters whenever (a) there is a significant mismatch between the
assumed model dynamics and the actual system, or (b) the posterior probability
tends to concentrate in relatively small regions of the state space. The
proposed scheme pushes some particles towards specific regions where the
likelihood is expected to be high, an operation known as nudging in the
geophysics literature. We re-interpret nudging in a form applicable to any
particle filtering scheme, as it does not involve any changes in the rest of
the algorithm. Since the particles are modified, but the importance weights do
not account for this modification, the use of nudging leads to additional bias
in the resulting estimators. However, we prove analytically that nudged
particle filters can still attain asymptotic convergence with the same error
rates as conventional particle methods. Simple analysis also yields an
alternative interpretation of the nudging operation that explains its
robustness to model errors. Finally, we show numerical results that illustrate
the improvements that can be attained using the proposed scheme. In particular,
we present nonlinear tracking examples with synthetic data and a model
inference example using real-world financial data
Adapting the Number of Particles in Sequential Monte Carlo Methods through an Online Scheme for Convergence Assessment
Particle filters are broadly used to approximate posterior distributions of
hidden states in state-space models by means of sets of weighted particles.
While the convergence of the filter is guaranteed when the number of particles
tends to infinity, the quality of the approximation is usually unknown but
strongly dependent on the number of particles. In this paper, we propose a
novel method for assessing the convergence of particle filters online manner,
as well as a simple scheme for the online adaptation of the number of particles
based on the convergence assessment. The method is based on a sequential
comparison between the actual observations and their predictive probability
distributions approximated by the filter. We provide a rigorous theoretical
analysis of the proposed methodology and, as an example of its practical use,
we present simulations of a simple algorithm for the dynamic and online
adaption of the number of particles during the operation of a particle filter
on a stochastic version of the Lorenz system
Variational semi-blind sparse deconvolution with orthogonal kernel bases and its application to MRFM
We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM)
Variational semi-blind sparse deconvolution with orthogonal kernel bases and its application to MRFM
We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM)
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