Input estimation and dynamical system identification

Recovery of signals from distorted or noisy observations has been a longstanding research problem with a wide variety of practical applications. We advocate to approach these types of problems by interpreting them as input estimation in finite-order linear state-space models. Among other applications the input signal may represent a physical quantity and the state-space model a sensor yielding corrupted readings. In this thesis, we provide new estimation algorithms and theoretical results for different classes of input signals: continuous-time input signals and weakly sparse input signals. The latter method is obtained by specializing a more general framework for inference with sparse priors and sparse signal recovery, which in contrast to standard methods, amounts to iterations of Gaussian message passing. Applicability of input estimation is extended to complex models, which generally are computationally more demanding and may be prone to numerical instability, by introducing new numerically robust computation methods expressed as Gaussian message passing in factor graphs. In practical applications, a signal model may not necessarily be available a-priori. As a consequence, in addition to input estimation, estimation of the state-space model itself must also be adressed. To this end, we introduce a variational statistical framework to retrieve convenient statespace models for input estimation and present a joint input and model estimation algorithm for weakly sparse input signals. The proposed methods are substantiated with two real world application examples. First, we consider impaired mechanical sensor measurements in machining processes and show that input estimation and suitable model identification can result in more accurate measurements, when strong resonances distort the sensor readings. Secondly, we show that our simultaneous weakly-sparse input estimation and model estimation method is capable of identifying individual heart beats from ballistocardiographic measurements, a method used to measure non-invasively the cardiac output.