  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.
