Linear recursive state estimators under uncertain observations

For linear systems with uncertain observations, we investigate the existence of recursive least-squares state estimators. The uncertainty in the observations is caused by a binary switching sequence γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , which is specified by a conditional probability distribution and which enters the observation equation as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zₖ = γₖ Hₖ xₖ+υₖ</tex> . Conditions are established which lead to a recursive filter for x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , and a procedure for constructing a mixture sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{γₖ}</tex> that satisfies these conditions is given. Such mixture sequences model the transmission of data in multichannels as in remote sensing situations as well as data links with random interruptions. They can also serve as models for communication in the presence of multiplicative noise.