Contributions to the filtering theory of continuous-time markovian jump linear systems

• Autor(a) principal: Vergés, Fortià Vila
• Data de Publicação: 2023
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações do LNCC
• Texto Completo: https://tede.lncc.br/handle/tede/366
• topic: Markov, Processos de; Sistemas lineares de controle; Sistemas dinâmicos lineares; Filtros (Matemática); Equações Diferenciais; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE

Stochastic differential equations with Markovian jump parameters constitute one of the most important class of hybrid dynamical systems, which has been extensively used for the modeling of dynamical systems which are subject to abrupt changes in their structure. The abrupt changes can be due, for instance, to abrupt environmental disturbances, component failure, volatility in economic systems, changes in subsystem interconnections, etc. This can be found, for instance, in aircraft control systems, robot systems, large flexible structures for the space station, etc. We shall be particularly interested in the class, which is dubbed in the literature as the class of Markov jump linear systems (MJLS) in which the jump mechanism is modeled by a Markov process, also known in the literature as the operation mode. We address the filtering problem for the operation mode in three different scenarios: (1) when the operation mode is detected via a noisy observation (the so-called hidden Markov processes); (2) the MJLS when the system signal is observable but not the operation mode, and (3) the MJLS when neither the system signal nor the operation mode is observable. For the first two scenarios, there exist in the literature finite optimal non-linear filter and infinite for the third. The main hindrances with the non-linear filter results are: (i) the non-linear filter performance depends heavily upon the stochastic numerical method used; (ii) it is not possible to devise a stationary version of the non-linear filters; and (iii) in the context of the control problem with partial observation of operation mode it introduces a great deal of nonlinearity in the Hamilton-Jacobi-Belman equation, which makes it difficult to get an explicit closed solution for the control problem. Motivated in part by this, the main contribution of this thesis is to devise the optimal linear filter for the operation mode for all the above mentioned scenarios. Besides, via the convergence study of the solution of a certain Riccati differential equation, to derive the associated stationary filter. In addition, relying on Murayamas stochastic numerical method and the results of Yuan and Mao, we carry out and analyze exhaustive simulations of all the filters devised in the thesis to illustrate their performance.