Discrete-time jump linear systems with Markov chain in a general state space.

• Autor(a) principal: Figueiredo, Danilo Zucolli
• Data de Publicação: 2016
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/3/3139/tde-18012017-115659/
• topic: Cadeias de Markov; Controle estocástico; Controle ótimo; Equações de Riccati; Linear systems; Markov chains; Optimal control; Riccati equations; Sistemas lineares; Stochastic control

This thesis deals with discrete-time Markov jump linear systems (MJLS) with Markov chain in a general Borel space S. Several control issues have been addressed for this class of dynamic systems, including stochastic stability (SS), linear quadratic (LQ) optimal control synthesis, fllter design and a separation principle. Necessary and sffcient conditions for SS have been derived. It was shown that SS is equivalent to the spectral radius of an operator being less than 1 or to the existence of a solution to a \\Lyapunov-like\" equation. Based on the SS concept, the finite- and infinite-horizon LQ optimal control problems were tackled. The solution to the finite- (infinite-)horizon LQ optimal control problem was derived from the associated control S-coupled Riccati difference (algebraic) equations. By S-coupled it is meant that the equations are coupled via an integral over a transition probability kernel having a density with respect to a in-finite measure on the Borel space S. The design of linear Markov jump filters was analyzed and a solution to the finite- (infinite-)horizon filtering problem was obtained based on the associated filtering S-coupled Riccati difference (algebraic) equations. Conditions for the existence and uniqueness of a stabilizing positive semi-definite solution to the control and filtering S-coupled algebraic Riccati equations have also been derived. Finally a separation principle for discrete-time MJLS with Markov chain in a general state space was obtained. It was shown that the optimal controller for a partial information optimal control problem separates the partial information control problem into two problems, one associated with a filtering problem and the other associated with an optimal control problem with complete information. It is expected that the results obtained in this thesis may motivate further research on discrete-time MJLS with Markov chain in a general state space.