Uniform Detectability-Aided Boundedness Analysis of Error Covariances of Kalman Filter for Time-Varying Systems

Some new results about the uniform lower and upper bounds of the predicted and filtered error covariances of the Kalman filter for linear time-varying systems from the perspective of uniform detectability are provided. First, a novel and general extendedly uniform detectability (EUD) is proposed, and the appropriate conditions to guarantee the existence of the uniform upper and lower bounds of the predicted and filtered error covariances are presented, respectively. Next, the explicit expressions of both upper and lower bounds are obtained with less conservatism. Moreover, the relations between those bounds, and their connections to estimation errors are revealed. Finally, an illustrative example is given to validate the effectiveness of the proposed theoretical developments in deriving the uniform lower and upper bounds, and a comparison with the existing results is made.