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Estimation of Dynamical Systems in Noisy Conditions and with Constraints

When measurements from dynamical systems are noisy, it is useful to have estimation algorithms that have low sensitivity to measurement noises and outliers. In the first set of results described in this paper we obtain optimal estimators for linear dynamical systems with $\epsilon$ insensitive loss functions. The $\epsilon$ insensitive loss function, which is often used in Support Vector Machines, provides greater robustness when the measurements are biased and very noisy as the algorithm tolerates small errors in prediction which in turn makes the estimates less sensitive to measurement noises. Apart from $\epsilon$ insensitive quadratic loss function, estimation algorithms are also derived for $\epsilon$ insensitive Huber M loss function which provides robustness in presence of both small noises as well as outliers. Robustness in presence of outliers is achieved with Huber cost function based estimator as the error penalty function switches from quadratic to linear for errors beyond certain threshold. The second set of results in the paper describe algorithms for estimation when apart from general description of dynamics of the system, one also has additional information about states and exogenous signals such as known range of some states or prior information about the maximum magnitude of noises/disturbances. While the proposed approaches have similarities to Kalman-Bucy or $\mathcal{H}_2$ smoothing algorithm, the algorithms are not linear in measurements but are easily implemented as optimal estimates are obtained by solving a standard quadratic optimization problem with linear constraints. For all cases, algorithms are proposed not only for filtering and smoothing but also for prediction of future states.

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