Discrete-Time Risk-Sensitive Filters with Non-Gaussian Initial Conditions and their Ergodic Properties

Abstract

In this paper, we study asymptotic stability properties ofrisk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant system with initial conditionsthat are not necessarily Gaussian. We show that in the case of Gaussianinitial conditions, the optimal risk-sensitive filter asymptoticallyconverges to any suboptimal filter initialized with an incorrect covariancematrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in arisk-sensitive filter that is asymptotically stable (that is, resultsin a solution for a Riccati equation that is asymptoticallystabilizing). For non-Gaussian initial conditions, we derive theexpression for the risk-sensitive filter in terms of a finite number ofparameters. Under a boundedness assumption satisfied by thefourth order moments of the initial state variable and a slow growthcondition satified by a certainRadon-Nikodym derivative, we show that a suboptimal risk-sensitive filterinitialized with Gaussian initial conditions asymptotically approachesthe optimal risk-sensitive filter for non-Gaussian initial conditions inthe mean square sense.The research and scientific content in this material has been submitted to the 1999 American Control Conference, San Diego, June 1999.