On the Stability of a Fixed Lag Smoother

The first. fixed lag smoothing algorithm mas developed by Rauch [I] as an extension to the fixed interval smoothing algorithm 121. Meditch [3] rederived t.he result,s of [I] using orthogonal project,ion. He conjectured that the h e d lag smoot.her [l], [3] would be asymptotically stable under the usual condit.ions which guarantee asymptotic stability of the Kalman filter. However, Kailath and Frost [4], while deriving t.he continuom-t,ime form of the smoothing equat.ions, pointed out that the equations representing the smoother and filter form a pair of adjoint systems. This would imply that the above conjecture about the stability of the fixed lag smoot.hing algorithm is false, as has been eslablished by Kelly and Anderson [5]. These aut,hors have shown that the fixed lag smoothing algorit.hm in [l], [3], and 141 are unstable, precisely when the corresponding filter is asymptotically stable. Because of the usefulness of thE form of smoothing in providing an on-line estimat,e that is superior to the filt,ered estimat.e, it is desirable to explore t,he possibility of deriving stable solutions of t.he smoothing problem. Two such solutions have recently been proposed [6], [i]. The purpose of this note is to show t,hat t.be smoothing algorithm developed earlier by t.he present authors [8] is asymptot.ically stable as long as the corresponding filter is asymptotically st,able. The stability is guaranteed by the fact that t,he structure of the smoot.her developed is analogous to that of the filter ikelf. For the sake of completeness, t.he smoother is derived here for the case of white Gaussian noise. The system is represented by the pair of equations