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When measurements from dynamical systems are noisy, it is useful to haveestimation algorithms that have low sensitivity to measurement noises andoutliers. In the first set of results described in this paper we obtain optimalestimators for linear dynamical systems with $\epsilon$ insensitive lossfunctions. The $\epsilon$ insensitive loss function, which is often used inSupport Vector Machines, provides greater robustness when the measurements arebiased and very noisy as the algorithm tolerates small errors in predictionwhich in turn makes the estimates less sensitive to measurement noises. Apartfrom $\epsilon$ insensitive quadratic loss function, estimation algorithms arealso derived for $\epsilon$ insensitive Huber M loss function which providesrobustness in presence of both small noises as well as outliers. Robustness inpresence of outliers is achieved with Huber cost function based estimator asthe error penalty function switches from quadratic to linear for errors beyondcertain threshold.  The second set of results in the paper describe algorithms for estimationwhen apart from general description of dynamics of the system, one also hasadditional information about states and exogenous signals such as known rangeof some states or prior information about the maximum magnitude ofnoises/disturbances. While the proposed approaches have similarities toKalman-Bucy or $\mathcal{H}_2$ smoothing algorithm, the algorithms are notlinear in measurements but are easily implemented as optimal estimates areobtained by solving a standard quadratic optimization problem with linearconstraints. For all cases, algorithms are proposed not only for filtering andsmoothing but also for prediction of future states.

Estimation of Dynamical Systems in Noisy Conditions and with Constraints