Analysis of reduced-search BCJR algorithms for input estimation in a jump linear system

Linear systems with unknown finite-valued inputs are of interest in all those hybrid frameworks where switches or jumps may change the continuous dynamics of a linear system. Many models have been proposed in this sense; in most cases, a probabilistic distribution on the input is assumed to be known and used as prior information for estimation. In this paper, we propose a simple model of jump linear system and develop low complexity algorithms, based on BCJR, to retrieve the input. We consider systems over a possibly infinite time horizon, which motivates the study of on-line, causal algorithms. Our main purpose is to provide a rigorous theoretical analysis of the performance of the proposed techniques: an error function is defined and its distribution is proved to converge, exploiting mathematical tools from Markov Processes theory and ergodic theorems.