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Discrete-time complementary models and smoothing. (English) Zbl 0676.93065

Summary: The fact that the smoothing error is a (wide sense) Markov process is somehow surprising since smoothed estimates depend upon both past and future data. In this paper we first give a simple and general proof of this fact. Then we use the so-called complementary models introduced by H. L. Weinert and U. D. Desai [IEEE Trans. Autom. Control AC- 26, 863-867 (1981; Zbl 0573.93074)] to derive forwards and backwards markovian models for the smoothing error in state-space models. By exploring the structure of the complementary models we show that, under certain restrictions, only two simple structured models exist, one that runs forwards in time and the other that runs backwards in time.
The forwards complementary model leads to the forward Rauch-Tung-Striebel (RTS) smoothing formula [H. Rauch, F. Tung and C. Striebel, AIAA J. 3, 1445-1450 (1965)] and to a backwards markovian model for the error, whereas the backwards model leads to the backward RTS formula and to a forwards error model.
The two models for the smoothing error can be derived one from the other by a forward-backward transformation that preserves the sample paths. Finally, by using a combination of the two complementary models we give yet another proof for the two-filter smoothing formula.

MSC:

93E14 Data smoothing in stochastic control theory
93C55 Discrete-time control/observation systems

Citations:

Zbl 0573.93074
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