Identification of optimum filter steady-state gain for systems with unknown noise covariances

A discrete linear stationary system is considered for which the input noise covariance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and the output noise covariance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> are unknown. A stable filter with a suboptimal gain is assumed. An identification scheme is presented which uses the autocorrelation functions of the innovations sequence of the suboptimal filter to determine the optimum filter steady state gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Γ</tex> directly without the intermediate determination of the unknown covariances <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> . The approach used is to identify an output equivalent representation of the original system which does not involve the unknown covariances directly.