Steady-state covariance analysis for a forward-pass fixed-interval smoother

This paper studies the solution of the steady-state error covariance equation (which is represented by the algebraic Lyapunov equation) associated with a forward-pass fixed-interval smoother for discrete-time linear systems. A necessary and sufficient condition is given to assure the existence of a unique stabilizing solution. A simple algorithm for solving such an equation is also proposed by using four eigenvector matrices, which are generated by a symplectic matrix, corresponding to the algebraic Riccati equation of a backward-pass information filter. Thus the results have application to the important problem of the limiting covariance analysis of smoothing prior to practically dealing with a finite interval of data.