New estimators for mixed stochastic and set theoretic uncertainty models: the scalar measurement case

Filters are derived for estimating the state of a linear dynamic system based on uncertain observations, which suffer from two types of uncertainties simultaneously. The first uncertainty is a stochastic process with given distribution. The second uncertainty is only known to be bounded, the exact underlying distribution is unknown. The new estimators combine set theoretic and stochastic estimation in a rigorous manner and provide a continuous transition between the two classical estimation concepts. They converge to a set theoretic estimator, when the stochastic error goes to zero, and to a Kalman filter, when the bounded error vanishes. In the mixed noise case, solution sets are provided that are uncertain in a stochastic sense.