Recursive Estimation for Dynamical Systems with Measurement Bias, Outliers and Constraints

This paper describes recursive algorithms for state estimation of linear dynamical systems when measurements are noisy with unknown bias and/or outliers. For situations with noisy and biased measurements, algorithms are proposed that minimize $\epsilon$ insensitive loss function. In this approach which is often used in Support Vector Machines, small errors are ignored making the algorithm less sensitive to measurement bias. Apart from $\epsilon$ insensitive quadratic loss function, estimation algorithms are also presented for $\epsilon$ insensitive Huber M loss function which provides good performance in presence of both small noises as well as outliers. The advantage of Huber cost function based estimator in presence of outliers is due to the fact the error penalty function switches from quadratic to linear for errors beyond a certain threshold. For both objective functions, estimation algorithms are extended to cases when there are additional constraints on states and exogenous signals such as known range of some states or exogenous signals or measurement noises. Interestingly, the filtering algorithms are recursive and structurally similar to Kalman filter with the main difference being that the updates based on the new measurement ("innovation term") are based on solution of a quadratic optimization problem with linear constraints.