Optimal Linear and Stationary Filter for Hidden Phase-Type Semi-Markov Models

• 成果类型: Article
• 署名作者: Verges, Fortia V.; Fragoso, Marcelo D.; Costa, Oswaldo Luiz do Valle
• 署名单位: Laboratorio Nacional de Computacao Cientifica (LNCC); Universidade de Sao Paulo
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3638075
• 发表日期: 2026
• 关键词: MEAN-SQUARE FILTER; probabilistic functions; STABILITY; EQUATIONS

摘要

The purpose of this article is to derive an optimal linear filter and its associated stationary filter for hidden phase-type (PH) semi-Markov models. We formulate the PH semi-Markov model as an equivalent Markov process with two components, the first one representing the mode of the semi-Markov process, and the second one associated with the transient states (phases). It is worth mentioning that PH distributions can approximate (in the weak convergence sense) any distribution with nonzero density in (0, infinity) with any desired precision. First, we derive an optimal linear filter for the hidden PH semi-Markov model, using the equivalent Markov process. Then, we derive the stationary filter, which essentially boils down to proving the convergence of the error covariance matrix associated with a Riccati differential equation to the stationary solution of an algebraic Riccati equation, in a manner analogous to the classical linear case. The difficulty increases in this case because the system matrices associated with the Riccati equation are not detectable. We conclude by illustrating the effectiveness of the filters derived here through some simulations.